diff --git a/Lib/test/test_cext/extension.c b/Lib/test/test_cext/extension.c
index 8a0f40d56b1ee4..56f40b354c6913 100644
--- a/Lib/test/test_cext/extension.c
+++ b/Lib/test/test_cext/extension.c
@@ -15,9 +15,11 @@
 
 #ifdef TEST_INTERNAL_C_API
    // gh-135906: Check for compiler warnings in the internal C API.
-   // - Cython uses pycore_frame.h.
+   // - Cython uses pycore_critical_section.h, pycore_frame.h and
+   //   pycore_template.h.
    // - greenlet uses pycore_frame.h, pycore_interpframe_structs.h and
    //   pycore_interpframe.h.
+#  include "internal/pycore_critical_section.h"
 #  include "internal/pycore_frame.h"
 #  include "internal/pycore_gc.h"
 #  include "internal/pycore_interp.h"
@@ -25,6 +27,7 @@
 #  include "internal/pycore_interpframe_structs.h"
 #  include "internal/pycore_object.h"
 #  include "internal/pycore_pystate.h"
+#  include "internal/pycore_template.h"
 #endif
 
 #ifndef MODULE_NAME
diff --git a/Lib/test/test_cppext/__init__.py b/Lib/test/test_cppext/__init__.py
index 9013503995bdce..1fd01702f64029 100644
--- a/Lib/test/test_cppext/__init__.py
+++ b/Lib/test/test_cppext/__init__.py
@@ -4,6 +4,7 @@
 import shlex
 import shutil
 import subprocess
+import sys
 import unittest
 from test import support
 
@@ -27,9 +28,6 @@
 class BaseTests:
     TEST_INTERNAL_C_API = False
 
-    def test_build(self):
-        self.check_build('_testcppext')
-
     def check_build(self, extension_name, std=None, limited=False):
         venv_dir = 'env'
         with support.setup_venv_with_pip_setuptools(venv_dir) as python_exe:
@@ -91,6 +89,9 @@ def run_cmd(operation, cmd):
 
 
 class TestPublicCAPI(BaseTests, unittest.TestCase):
+    def test_build(self):
+        self.check_build('_testcppext')
+
     @support.requires_gil_enabled('incompatible with Free Threading')
     def test_build_limited_cpp03(self):
         self.check_build('_test_limited_cpp03ext', std='c++03', limited=True)
@@ -119,6 +120,13 @@ def test_build_cpp14(self):
 class TestInteralCAPI(BaseTests, unittest.TestCase):
     TEST_INTERNAL_C_API = True
 
+    def test_build(self):
+        kwargs = {}
+        if sys.platform == 'darwin':
+            # Old Apple clang++ default C++ std is gnu++98
+            kwargs['std'] = 'c++11'
+        self.check_build('_testcppext_internal', **kwargs)
+
 
 if __name__ == "__main__":
     unittest.main()
diff --git a/Lib/test/test_cppext/extension.cpp b/Lib/test/test_cppext/extension.cpp
index 811374c0361e70..4db63df94f5233 100644
--- a/Lib/test/test_cppext/extension.cpp
+++ b/Lib/test/test_cppext/extension.cpp
@@ -15,14 +15,20 @@
 
 #ifdef TEST_INTERNAL_C_API
    // gh-135906: Check for compiler warnings in the internal C API
+   // - Cython uses pycore_critical_section.h, pycore_frame.h and
+   //   pycore_template.h.
+   // - greenlet uses pycore_frame.h, pycore_interpframe_structs.h and
+   //   pycore_interpframe.h.
 #  include "internal/pycore_frame.h"
-   // mimalloc emits many compiler warnings when Python is built in debug
-   // mode (when MI_DEBUG is not zero).
-   // mimalloc emits compiler warnings when Python is built on Windows
-   // and macOS.
-#  if !defined(Py_DEBUG) && !defined(MS_WINDOWS) && !defined(__APPLE__)
+#  include "internal/pycore_interpframe_structs.h"
+#  include "internal/pycore_template.h"
+
+   // mimalloc emits compiler warnings on Windows.
+#  if !defined(MS_WINDOWS)
 #    include "internal/pycore_backoff.h"
 #    include "internal/pycore_cell.h"
+#    include "internal/pycore_critical_section.h"
+#    include "internal/pycore_interpframe.h"
 #  endif
 #endif
 
diff --git a/Lib/test/test_cppext/setup.py b/Lib/test/test_cppext/setup.py
index a3eec1c67e1556..2d9052a6b879da 100644
--- a/Lib/test/test_cppext/setup.py
+++ b/Lib/test/test_cppext/setup.py
@@ -59,7 +59,7 @@ def main():
         else:
             cppflags.append(f'-std={std}')
 
-        if limited or (std != 'c++03'):
+        if limited or (std != 'c++03') and not internal:
             # See CPPFLAGS_PEDANTIC docstring
             cppflags.extend(CPPFLAGS_PEDANTIC)