Reconcile computation of isolated build environment paths

Use the same code to determine isolated environment paths at
dependency install time and at environment setup time. We do not care
about the exact paths but the paths needs to be consistent at package
installation time and environment setup.

This should fix all issues observed on platforms that customize the
installation schemes, such as Debian and Homebrew, where dependency
installation and isolated build environment setup resolved to
different paths.

See pypa#11598 and pypa#11623.

news/11740.bugfix.rst

@@ -0,0 +1,4 @@
+Reconcile computation of isolated build environment paths. Fix all
+issues observed on platforms that customize the installation schemes,
+such as Debian and Homebrew, where dependency installation and
+isolated build environment setup resolved to different paths.

src/pip/_internal/build_env.py

@@ -9,20 +9,15 @@
 import textwrap
 from collections import OrderedDict
 from types import TracebackType
-from typing import TYPE_CHECKING, Iterable, List, Optional, Set, Tuple, Type
+from typing import TYPE_CHECKING, Iterable, List, Optional, Set, Tuple, Type, Union
 
 from pip._vendor.certifi import where
 from pip._vendor.packaging.requirements import Requirement
 from pip._vendor.packaging.version import Version
 
 from pip import __file__ as pip_location
 from pip._internal.cli.spinners import open_spinner
-from pip._internal.locations import (
-    get_isolated_environment_bin_path,
-    get_isolated_environment_lib_paths,
-    get_platlib,
-    get_purelib,
-)
+from pip._internal.locations import get_platlib, get_purelib, get_scheme
 from pip._internal.metadata import get_default_environment, get_environment
 from pip._internal.utils.subprocess import call_subprocess
 from pip._internal.utils.temp_dir import TempDirectory, tempdir_kinds
@@ -33,12 +28,17 @@
 logger = logging.getLogger(__name__)
 
 
+def _dedup(a: str, b: str) -> Union[Tuple[str], Tuple[str, str]]:
+    return (a, b) if a != b else (a,)
+
+
 class _Prefix:
     def __init__(self, path: str) -> None:
         self.path = path
         self.setup = False
-        self.bin_dir = get_isolated_environment_bin_path(path)
-        self.lib_dirs = get_isolated_environment_lib_paths(path)
+        scheme = get_scheme("", prefix=path)
+        self.bin_dir = scheme.scripts
+        self.lib_dirs = _dedup(scheme.purelib, scheme.platlib)
 
 
 def get_runnable_pip() -> str: