diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 4220061b2385b..e83bc9c1448eb 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -2580,7 +2580,6 @@ def __nonzero__(self):
     # --------------------------------------------------------------------
     # Set Operation Methods
 
-    @final
     def _get_reconciled_name_object(self, other):
         """
         If the result of a set operation will be self,
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index cb0ca335f751f..385b7e487d4c5 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -3593,6 +3593,34 @@ def _union(self, other, sort):
     def _is_comparable_dtype(self, dtype: DtypeObj) -> bool:
         return is_object_dtype(dtype)
 
+    def _get_reconciled_name_object(self, other):
+        """
+        If the result of a set operation will be self,
+        return self, unless the names change, in which
+        case make a shallow copy of self.
+        """
+        names = self._maybe_match_names(other)
+        if self.names != names:
+            return self.rename(names)
+        return self
+
+    def _maybe_match_names(self, other):
+        """
+        Try to find common names to attach to the result of an operation between
+        a and b.  Return a consensus list of names if they match at least partly
+        or None if they have completely different names.
+        """
+        if len(self.names) != len(other.names):
+            return None
+        names = []
+        for a_name, b_name in zip(self.names, other.names):
+            if a_name == b_name:
+                names.append(a_name)
+            else:
+                # TODO: what if they both have np.nan for their names?
+                names.append(None)
+        return names
+
     def intersection(self, other, sort=False):
         """
         Form the intersection of two MultiIndex objects.
@@ -3616,12 +3644,12 @@ def intersection(self, other, sort=False):
         """
         self._validate_sort_keyword(sort)
         self._assert_can_do_setop(other)
-        other, result_names = self._convert_can_do_setop(other)
+        other, _ = self._convert_can_do_setop(other)
 
         if self.equals(other):
             if self.has_duplicates:
-                return self.unique().rename(result_names)
-            return self.rename(result_names)
+                return self.unique()._get_reconciled_name_object(other)
+            return self._get_reconciled_name_object(other)
 
         return self._intersection(other, sort=sort)
 
diff --git a/pandas/core/ops/common.py b/pandas/core/ops/common.py
index a6bcab44e5519..58ad3237c8288 100644
--- a/pandas/core/ops/common.py
+++ b/pandas/core/ops/common.py
@@ -93,7 +93,7 @@ def _maybe_match_name(a, b):
     """
     Try to find a name to attach to the result of an operation between
     a and b.  If only one of these has a `name` attribute, return that
-    name.  Otherwise return a consensus name if they match of None if
+    name.  Otherwise return a consensus name if they match or None if
     they have different names.
 
     Parameters
diff --git a/pandas/tests/indexes/multi/test_setops.py b/pandas/tests/indexes/multi/test_setops.py
index f4f602c780187..f9fc425e46696 100644
--- a/pandas/tests/indexes/multi/test_setops.py
+++ b/pandas/tests/indexes/multi/test_setops.py
@@ -421,6 +421,29 @@ def test_intersect_with_duplicates(tuples, exp_tuples):
     tm.assert_index_equal(result, expected)
 
 
+@pytest.mark.parametrize(
+    "data, names, expected",
+    [
+        ((1,), None, None),
+        ((1,), ["a"], None),
+        ((1,), ["b"], None),
+        ((1, 2), ["c", "d"], [None, None]),
+        ((1, 2), ["b", "a"], [None, None]),
+        ((1, 2, 3), ["a", "b", "c"], None),
+        ((1, 2), ["a", "c"], ["a", None]),
+        ((1, 2), ["c", "b"], [None, "b"]),
+        ((1, 2), ["a", "b"], ["a", "b"]),
+        ((1, 2), [None, "b"], [None, "b"]),
+    ],
+)
+def test_maybe_match_names(data, names, expected):
+    # GH#38323
+    mi = pd.MultiIndex.from_tuples([], names=["a", "b"])
+    mi2 = pd.MultiIndex.from_tuples([data], names=names)
+    result = mi._maybe_match_names(mi2)
+    assert result == expected
+
+
 def test_intersection_equal_different_names():
     # GH#30302
     mi1 = MultiIndex.from_arrays([[1, 2], [3, 4]], names=["c", "b"])
@@ -429,3 +452,11 @@ def test_intersection_equal_different_names():
     result = mi1.intersection(mi2)
     expected = MultiIndex.from_arrays([[1, 2], [3, 4]], names=[None, "b"])
     tm.assert_index_equal(result, expected)
+
+
+def test_intersection_different_names():
+    # GH#38323
+    mi = MultiIndex.from_arrays([[1], [3]], names=["c", "b"])
+    mi2 = MultiIndex.from_arrays([[1], [3]])
+    result = mi.intersection(mi2)
+    tm.assert_index_equal(result, mi2)