gh-83856: Honor atexit for all multiprocessing start methods (GH-114279)

Use atexit for all multiprocessing start methods to cleanup.
See the GH-114279 PR discussion and related issue for details as to why.

Lib/test/_test_multiprocessing.py

@@ -6161,6 +6161,29 @@ def submain(): pass
         self.assertFalse(err, msg=err.decode('utf-8'))
 
 
+class _TestAtExit(BaseTestCase):
+
+    ALLOWED_TYPES = ('processes',)
+
+    @classmethod
+    def _write_file_at_exit(self, output_path):
+        import atexit
+        def exit_handler():
+            with open(output_path, 'w') as f:
+                f.write("deadbeef")
+        atexit.register(exit_handler)
+
+    def test_atexit(self):
+        # gh-83856
+        with os_helper.temp_dir() as temp_dir:
+            output_path = os.path.join(temp_dir, 'output.txt')
+            p = self.Process(target=self._write_file_at_exit, args=(output_path,))
+            p.start()
+            p.join()
+            with open(output_path) as f:
+                self.assertEqual(f.read(), 'deadbeef')
+
+
 class MiscTestCase(unittest.TestCase):
     def test__all__(self):
         # Just make sure names in not_exported are excluded