diff --git a/Lib/multiprocessing/process.py b/Lib/multiprocessing/process.py
--- a/Lib/multiprocessing/process.py
+++ b/Lib/multiprocessing/process.py
@@ -281,8 +281,8 @@ class Process(object):
             exitcode = 1
             import traceback
             sys.stderr.write('Process %s:\n' % self.name)
+            traceback.print_exc()
             sys.stderr.flush()
-            traceback.print_exc()
 
         util.info('process exiting with exitcode %d' % exitcode)
         return exitcode
diff --git a/Lib/test/test_multiprocessing.py b/Lib/test/test_multiprocessing.py
--- a/Lib/test/test_multiprocessing.py
+++ b/Lib/test/test_multiprocessing.py
@@ -328,6 +328,28 @@ class _TestProcess(BaseTestCase):
             ]
         self.assertEqual(result, expected)
 
+    def test_stderr_flush(self):
+        # sys.stderr is flushed at process shutdown (issue #13812)
+        if self.TYPE == "threads":
+            return
+
+        testfn = test.support.TESTFN
+        proc = self.Process(target=self._test_stderr_flush, args=(testfn,))
+        proc.start()
+        proc.join()
+        with open(testfn, 'r') as f:
+            err = f.read()
+            # The whole traceback was printed
+            self.assertIn("ZeroDivisionError", err)
+            self.assertIn("test_multiprocessing.py", err)
+            self.assertIn("1/0 # MARKER", err)
+
+    @classmethod
+    def _test_stderr_flush(cls, testfn):
+        sys.stderr = open(testfn, 'w')
+        1/0 # MARKER
+
+
 #
 #
 #