diff --git a/CMakeLists.txt b/CMakeLists.txt
index 25a7d14984..0ad74db2bb 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -126,6 +126,7 @@ set(PYBIND11_HEADERS
     include/pybind11/complex.h
     include/pybind11/options.h
     include/pybind11/eigen.h
+    include/pybind11/eigen/common.h
     include/pybind11/eigen/matrix.h
     include/pybind11/eigen/tensor.h
     include/pybind11/embed.h
diff --git a/include/pybind11/buffer_info.h b/include/pybind11/buffer_info.h
index 06120d5563..b99ee8bef4 100644
--- a/include/pybind11/buffer_info.h
+++ b/include/pybind11/buffer_info.h
@@ -37,6 +37,9 @@ inline std::vector<ssize_t> f_strides(const std::vector<ssize_t> &shape, ssize_t
     return strides;
 }
 
+template <typename T, typename SFINAE = void>
+struct compare_buffer_info;
+
 PYBIND11_NAMESPACE_END(detail)
 
 /// Information record describing a Python buffer object
@@ -150,6 +153,17 @@ struct buffer_info {
     Py_buffer *view() const { return m_view; }
     Py_buffer *&view() { return m_view; }
 
+    /* True if the buffer item type is equivalent to `T`. */
+    // To define "equivalent" by example:
+    // `buffer_info::item_type_is_equivalent_to<int>(b)` and
+    // `buffer_info::item_type_is_equivalent_to<long>(b)` may both be true
+    // on some platforms, but `int` and `unsigned` will never be equivalent.
+    // For the ground truth, please inspect `detail::compare_buffer_info<>`.
+    template <typename T>
+    bool item_type_is_equivalent_to() const {
+        return detail::compare_buffer_info<T>::compare(*this);
+    }
+
 private:
     struct private_ctr_tag {};
 
@@ -170,9 +184,10 @@ struct buffer_info {
 
 PYBIND11_NAMESPACE_BEGIN(detail)
 
-template <typename T, typename SFINAE = void>
+template <typename T, typename SFINAE>
 struct compare_buffer_info {
     static bool compare(const buffer_info &b) {
+        // NOLINTNEXTLINE(bugprone-sizeof-expression) Needed for `PyObject *`
         return b.format == format_descriptor<T>::format() && b.itemsize == (ssize_t) sizeof(T);
     }
 };
diff --git a/include/pybind11/detail/common.h b/include/pybind11/detail/common.h
index 1ff0df09fe..c2e6f9ec89 100644
--- a/include/pybind11/detail/common.h
+++ b/include/pybind11/detail/common.h
@@ -1025,6 +1025,15 @@ PYBIND11_RUNTIME_EXCEPTION(reference_cast_error, PyExc_RuntimeError) /// Used in
 template <typename T, typename SFINAE = void>
 struct format_descriptor {};
 
+template <typename T>
+struct format_descriptor<
+    T,
+    detail::enable_if_t<detail::is_same_ignoring_cvref<T, PyObject *>::value>> {
+    static constexpr const char c = 'O';
+    static constexpr const char value[2] = {c, '\0'};
+    static std::string format() { return std::string(1, c); }
+};
+
 PYBIND11_NAMESPACE_BEGIN(detail)
 // Returns the index of the given type in the type char array below, and in the list in numpy.h
 // The order here is: bool; 8 ints ((signed,unsigned)x(8,16,32,64)bits); float,double,long double;
diff --git a/include/pybind11/eigen/common.h b/include/pybind11/eigen/common.h
new file mode 100644
index 0000000000..24f56d1584
--- /dev/null
+++ b/include/pybind11/eigen/common.h
@@ -0,0 +1,9 @@
+// Copyright (c) 2023 The pybind Community.
+
+#pragma once
+
+// Common message for `static_assert()`s, which are useful to easily
+// preempt much less obvious errors.
+#define PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED                                    \
+    "Pointer types (in particular `PyObject *`) are not supported as scalar types for Eigen "     \
+    "types."
diff --git a/include/pybind11/eigen/matrix.h b/include/pybind11/eigen/matrix.h
index bd533bed37..8d4342f81b 100644
--- a/include/pybind11/eigen/matrix.h
+++ b/include/pybind11/eigen/matrix.h
@@ -10,6 +10,7 @@
 #pragma once
 
 #include "../numpy.h"
+#include "common.h"
 
 /* HINT: To suppress warnings originating from the Eigen headers, use -isystem.
    See also:
@@ -287,6 +288,8 @@ handle eigen_encapsulate(Type *src) {
 template <typename Type>
 struct type_caster<Type, enable_if_t<is_eigen_dense_plain<Type>::value>> {
     using Scalar = typename Type::Scalar;
+    static_assert(!std::is_pointer<Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
     using props = EigenProps<Type>;
 
     bool load(handle src, bool convert) {
@@ -405,6 +408,9 @@ struct type_caster<Type, enable_if_t<is_eigen_dense_plain<Type>::value>> {
 // Base class for casting reference/map/block/etc. objects back to python.
 template <typename MapType>
 struct eigen_map_caster {
+    static_assert(!std::is_pointer<typename MapType::Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
+
 private:
     using props = EigenProps<MapType>;
 
@@ -457,6 +463,8 @@ struct type_caster<
     using Type = Eigen::Ref<PlainObjectType, 0, StrideType>;
     using props = EigenProps<Type>;
     using Scalar = typename props::Scalar;
+    static_assert(!std::is_pointer<Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
     using MapType = Eigen::Map<PlainObjectType, 0, StrideType>;
     using Array
         = array_t<Scalar,
@@ -604,6 +612,9 @@ struct type_caster<
 // regular Eigen::Matrix, then casting that.
 template <typename Type>
 struct type_caster<Type, enable_if_t<is_eigen_other<Type>::value>> {
+    static_assert(!std::is_pointer<typename Type::Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
+
 protected:
     using Matrix
         = Eigen::Matrix<typename Type::Scalar, Type::RowsAtCompileTime, Type::ColsAtCompileTime>;
@@ -632,6 +643,8 @@ struct type_caster<Type, enable_if_t<is_eigen_other<Type>::value>> {
 template <typename Type>
 struct type_caster<Type, enable_if_t<is_eigen_sparse<Type>::value>> {
     using Scalar = typename Type::Scalar;
+    static_assert(!std::is_pointer<Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
     using StorageIndex = remove_reference_t<decltype(*std::declval<Type>().outerIndexPtr())>;
     using Index = typename Type::Index;
     static constexpr bool rowMajor = Type::IsRowMajor;
diff --git a/include/pybind11/eigen/tensor.h b/include/pybind11/eigen/tensor.h
index de7dcba89d..25d12baca1 100644
--- a/include/pybind11/eigen/tensor.h
+++ b/include/pybind11/eigen/tensor.h
@@ -8,6 +8,7 @@
 #pragma once
 
 #include "../numpy.h"
+#include "common.h"
 
 #if defined(__GNUC__) && !defined(__clang__) && !defined(__INTEL_COMPILER)
 static_assert(__GNUC__ > 5, "Eigen Tensor support in pybind11 requires GCC > 5.0");
@@ -164,6 +165,8 @@ PYBIND11_WARNING_POP
 
 template <typename Type>
 struct type_caster<Type, typename eigen_tensor_helper<Type>::ValidType> {
+    static_assert(!std::is_pointer<typename Type::Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
     using Helper = eigen_tensor_helper<Type>;
     static constexpr auto temp_name = get_tensor_descriptor<Type, false>::value;
     PYBIND11_TYPE_CASTER(Type, temp_name);
@@ -359,6 +362,8 @@ struct get_storage_pointer_type<MapType, void_t<typename MapType::PointerArgType
 template <typename Type, int Options>
 struct type_caster<Eigen::TensorMap<Type, Options>,
                    typename eigen_tensor_helper<remove_cv_t<Type>>::ValidType> {
+    static_assert(!std::is_pointer<typename Type::Scalar>::value,
+                  PYBIND11_EIGEN_MESSAGE_POINTER_TYPES_ARE_NOT_SUPPORTED);
     using MapType = Eigen::TensorMap<Type, Options>;
     using Helper = eigen_tensor_helper<remove_cv_t<Type>>;
 
diff --git a/include/pybind11/numpy.h b/include/pybind11/numpy.h
index 854d6e87fa..36077ec04d 100644
--- a/include/pybind11/numpy.h
+++ b/include/pybind11/numpy.h
@@ -564,6 +564,8 @@ class dtype : public object {
         m_ptr = from_args(args).release().ptr();
     }
 
+    /// Return dtype for the given typenum (one of the NPY_TYPES).
+    /// https://numpy.org/devdocs/reference/c-api/array.html#c.PyArray_DescrFromType
     explicit dtype(int typenum)
         : object(detail::npy_api::get().PyArray_DescrFromType_(typenum), stolen_t{}) {
         if (m_ptr == nullptr) {
@@ -1283,12 +1285,16 @@ struct npy_format_descriptor<
 public:
     static constexpr int value = values[detail::is_fmt_numeric<T>::index];
 
-    static pybind11::dtype dtype() {
-        if (auto *ptr = npy_api::get().PyArray_DescrFromType_(value)) {
-            return reinterpret_steal<pybind11::dtype>(ptr);
-        }
-        pybind11_fail("Unsupported buffer format!");
-    }
+    static pybind11::dtype dtype() { return pybind11::dtype(/*typenum*/ value); }
+};
+
+template <typename T>
+struct npy_format_descriptor<T, enable_if_t<is_same_ignoring_cvref<T, PyObject *>::value>> {
+    static constexpr auto name = const_name("object");
+
+    static constexpr int value = npy_api::NPY_OBJECT_;
+
+    static pybind11::dtype dtype() { return pybind11::dtype(/*typenum*/ value); }
 };
 
 #define PYBIND11_DECL_CHAR_FMT                                                                    \
diff --git a/tests/extra_python_package/test_files.py b/tests/extra_python_package/test_files.py
index e5982f962e..f3916ee1f9 100644
--- a/tests/extra_python_package/test_files.py
+++ b/tests/extra_python_package/test_files.py
@@ -57,6 +57,7 @@
 }
 
 eigen_headers = {
+    "include/pybind11/eigen/common.h",
     "include/pybind11/eigen/matrix.h",
     "include/pybind11/eigen/tensor.h",
 }
diff --git a/tests/test_buffers.cpp b/tests/test_buffers.cpp
index 6b6e8cba7f..b5b8c355b3 100644
--- a/tests/test_buffers.cpp
+++ b/tests/test_buffers.cpp
@@ -7,12 +7,47 @@
     BSD-style license that can be found in the LICENSE file.
 */
 
+#include <pybind11/complex.h>
 #include <pybind11/stl.h>
 
 #include "constructor_stats.h"
 #include "pybind11_tests.h"
 
 TEST_SUBMODULE(buffers, m) {
+    m.attr("long_double_and_double_have_same_size") = (sizeof(long double) == sizeof(double));
+
+    m.def("format_descriptor_format_buffer_info_equiv",
+          [](const std::string &cpp_name, const py::buffer &buffer) {
+              // https://google.github.io/styleguide/cppguide.html#Static_and_Global_Variables
+              static auto *format_table = new std::map<std::string, std::string>;
+              static auto *equiv_table
+                  = new std::map<std::string, bool (py::buffer_info::*)() const>;
+              if (format_table->empty()) {
+#define PYBIND11_ASSIGN_HELPER(...)                                                               \
+    (*format_table)[#__VA_ARGS__] = py::format_descriptor<__VA_ARGS__>::format();                 \
+    (*equiv_table)[#__VA_ARGS__] = &py::buffer_info::item_type_is_equivalent_to<__VA_ARGS__>;
+                  PYBIND11_ASSIGN_HELPER(PyObject *)
+                  PYBIND11_ASSIGN_HELPER(bool)
+                  PYBIND11_ASSIGN_HELPER(std::int8_t)
+                  PYBIND11_ASSIGN_HELPER(std::uint8_t)
+                  PYBIND11_ASSIGN_HELPER(std::int16_t)
+                  PYBIND11_ASSIGN_HELPER(std::uint16_t)
+                  PYBIND11_ASSIGN_HELPER(std::int32_t)
+                  PYBIND11_ASSIGN_HELPER(std::uint32_t)
+                  PYBIND11_ASSIGN_HELPER(std::int64_t)
+                  PYBIND11_ASSIGN_HELPER(std::uint64_t)
+                  PYBIND11_ASSIGN_HELPER(float)
+                  PYBIND11_ASSIGN_HELPER(double)
+                  PYBIND11_ASSIGN_HELPER(long double)
+                  PYBIND11_ASSIGN_HELPER(std::complex<float>)
+                  PYBIND11_ASSIGN_HELPER(std::complex<double>)
+                  PYBIND11_ASSIGN_HELPER(std::complex<long double>)
+#undef PYBIND11_ASSIGN_HELPER
+              }
+              return std::pair<std::string, bool>(
+                  (*format_table)[cpp_name], (buffer.request().*((*equiv_table)[cpp_name]))());
+          });
+
     // test_from_python / test_to_python:
     class Matrix {
     public:
diff --git a/tests/test_buffers.py b/tests/test_buffers.py
index eb58c4675e..63d9d869fd 100644
--- a/tests/test_buffers.py
+++ b/tests/test_buffers.py
@@ -10,6 +10,63 @@
 
 np = pytest.importorskip("numpy")
 
+if m.long_double_and_double_have_same_size:
+    # Determined by the compiler used to build the pybind11 tests
+    # (e.g. MSVC gets here, but MinGW might not).
+    np_float128 = None
+    np_complex256 = None
+else:
+    # Determined by the compiler used to build numpy (e.g. MinGW).
+    np_float128 = getattr(np, *["float128"] * 2)
+    np_complex256 = getattr(np, *["complex256"] * 2)
+
+CPP_NAME_FORMAT_NP_DTYPE_TABLE = [
+    ("PyObject *", "O", object),
+    ("bool", "?", np.bool_),
+    ("std::int8_t", "b", np.int8),
+    ("std::uint8_t", "B", np.uint8),
+    ("std::int16_t", "h", np.int16),
+    ("std::uint16_t", "H", np.uint16),
+    ("std::int32_t", "i", np.int32),
+    ("std::uint32_t", "I", np.uint32),
+    ("std::int64_t", "q", np.int64),
+    ("std::uint64_t", "Q", np.uint64),
+    ("float", "f", np.float32),
+    ("double", "d", np.float64),
+    ("long double", "g", np_float128),
+    ("std::complex<float>", "Zf", np.complex64),
+    ("std::complex<double>", "Zd", np.complex128),
+    ("std::complex<long double>", "Zg", np_complex256),
+]
+CPP_NAME_FORMAT_TABLE = [
+    (cpp_name, format)
+    for cpp_name, format, np_dtype in CPP_NAME_FORMAT_NP_DTYPE_TABLE
+    if np_dtype is not None
+]
+CPP_NAME_NP_DTYPE_TABLE = [
+    (cpp_name, np_dtype) for cpp_name, _, np_dtype in CPP_NAME_FORMAT_NP_DTYPE_TABLE
+]
+
+
+@pytest.mark.parametrize(("cpp_name", "np_dtype"), CPP_NAME_NP_DTYPE_TABLE)
+def test_format_descriptor_format_buffer_info_equiv(cpp_name, np_dtype):
+    if np_dtype is None:
+        pytest.skip(
+            f"cpp_name=`{cpp_name}`: `long double` and `double` have same size."
+        )
+    if isinstance(np_dtype, str):
+        pytest.skip(f"np.{np_dtype} does not exist.")
+    np_array = np.array([], dtype=np_dtype)
+    for other_cpp_name, expected_format in CPP_NAME_FORMAT_TABLE:
+        format, np_array_is_matching = m.format_descriptor_format_buffer_info_equiv(
+            other_cpp_name, np_array
+        )
+        assert format == expected_format
+        if other_cpp_name == cpp_name:
+            assert np_array_is_matching
+        else:
+            assert not np_array_is_matching
+
 
 def test_from_python():
     with pytest.raises(RuntimeError) as excinfo:
diff --git a/tests/test_numpy_array.cpp b/tests/test_numpy_array.cpp
index b118e2c6cc..8c122a8658 100644
--- a/tests/test_numpy_array.cpp
+++ b/tests/test_numpy_array.cpp
@@ -523,4 +523,30 @@ TEST_SUBMODULE(numpy_array, sm) {
     sm.def("test_fmt_desc_const_double", [](const py::array_t<const double> &) {});
 
     sm.def("round_trip_float", [](double d) { return d; });
+
+    sm.def("pass_array_pyobject_ptr_return_sum_str_values",
+           [](const py::array_t<PyObject *> &objs) {
+               std::string sum_str_values;
+               for (const auto &obj : objs) {
+                   sum_str_values += py::str(obj.attr("value"));
+               }
+               return sum_str_values;
+           });
+
+    sm.def("pass_array_pyobject_ptr_return_as_list",
+           [](const py::array_t<PyObject *> &objs) -> py::list { return objs; });
+
+    sm.def("return_array_pyobject_ptr_cpp_loop", [](const py::list &objs) {
+        py::size_t arr_size = py::len(objs);
+        py::array_t<PyObject *> arr_from_list(static_cast<py::ssize_t>(arr_size));
+        PyObject **data = arr_from_list.mutable_data();
+        for (py::size_t i = 0; i < arr_size; i++) {
+            assert(data[i] == nullptr);
+            data[i] = py::cast<PyObject *>(objs[i].attr("value"));
+        }
+        return arr_from_list;
+    });
+
+    sm.def("return_array_pyobject_ptr_from_list",
+           [](const py::list &objs) -> py::array_t<PyObject *> { return objs; });
 }
diff --git a/tests/test_numpy_array.py b/tests/test_numpy_array.py
index 070813d3a2..12e7d17d15 100644
--- a/tests/test_numpy_array.py
+++ b/tests/test_numpy_array.py
@@ -595,3 +595,74 @@ def test_round_trip_float():
     arr = np.zeros((), np.float64)
     arr[()] = 37.2
     assert m.round_trip_float(arr) == 37.2
+
+
+# HINT: An easy and robust way (although only manual unfortunately) to check for
+#       ref-count leaks in the test_.*pyobject_ptr.* functions below is to
+#           * temporarily insert `while True:` (one-by-one),
+#           * run this test, and
+#           * run the Linux `top` command in another shell to visually monitor
+#             `RES` for a minute or two.
+#       If there is a leak, it is usually evident in seconds because the `RES`
+#       value increases without bounds. (Don't forget to Ctrl-C the test!)
+
+
+# For use as a temporary user-defined object, to maximize sensitivity of the tests below:
+#     * Ref-count leaks will be immediately evident.
+#     * Sanitizers are much more likely to detect heap-use-after-free due to
+#       other ref-count bugs.
+class PyValueHolder:
+    def __init__(self, value):
+        self.value = value
+
+
+def WrapWithPyValueHolder(*values):
+    return [PyValueHolder(v) for v in values]
+
+
+def UnwrapPyValueHolder(vhs):
+    return [vh.value for vh in vhs]
+
+
+def test_pass_array_pyobject_ptr_return_sum_str_values_ndarray():
+    # Intentionally all temporaries, do not change.
+    assert (
+        m.pass_array_pyobject_ptr_return_sum_str_values(
+            np.array(WrapWithPyValueHolder(-3, "four", 5.0), dtype=object)
+        )
+        == "-3four5.0"
+    )
+
+
+def test_pass_array_pyobject_ptr_return_sum_str_values_list():
+    # Intentionally all temporaries, do not change.
+    assert (
+        m.pass_array_pyobject_ptr_return_sum_str_values(
+            WrapWithPyValueHolder(2, "three", -4.0)
+        )
+        == "2three-4.0"
+    )
+
+
+def test_pass_array_pyobject_ptr_return_as_list():
+    # Intentionally all temporaries, do not change.
+    assert UnwrapPyValueHolder(
+        m.pass_array_pyobject_ptr_return_as_list(
+            np.array(WrapWithPyValueHolder(-1, "two", 3.0), dtype=object)
+        )
+    ) == [-1, "two", 3.0]
+
+
+@pytest.mark.parametrize(
+    ("return_array_pyobject_ptr", "unwrap"),
+    [
+        (m.return_array_pyobject_ptr_cpp_loop, list),
+        (m.return_array_pyobject_ptr_from_list, UnwrapPyValueHolder),
+    ],
+)
+def test_return_array_pyobject_ptr_cpp_loop(return_array_pyobject_ptr, unwrap):
+    # Intentionally all temporaries, do not change.
+    arr_from_list = return_array_pyobject_ptr(WrapWithPyValueHolder(6, "seven", -8.0))
+    assert isinstance(arr_from_list, np.ndarray)
+    assert arr_from_list.dtype == np.dtype("O")
+    assert unwrap(arr_from_list) == [6, "seven", -8.0]