diff --git a/Lib/unittest/mock.py b/Lib/unittest/mock.py
index a98decc..513cd5c 100644
--- a/Lib/unittest/mock.py
+++ b/Lib/unittest/mock.py
@@ -852,7 +852,8 @@
             else:
                 name, args, kwargs = _call
             try:
-                return name, sig.bind(*args, **kwargs)
+                bound_call = sig.bind(*args, **kwargs)
+                return call(name, bound_call.args, bound_call.kwargs)
             except TypeError as e:
                 return e.with_traceback(None)
         else:
@@ -901,9 +902,9 @@
         def _error_message():
             msg = self._format_mock_failure_message(args, kwargs)
             return msg
-        expected = self._call_matcher((args, kwargs))
+        expected = self._call_matcher(_Call((args, kwargs), two=True))
         actual = self._call_matcher(self.call_args)
-        if expected != actual:
+        if actual != expected:
             cause = expected if isinstance(expected, Exception) else None
             raise AssertionError(_error_message()) from cause
 
@@ -963,10 +964,10 @@
         The assert passes if the mock has *ever* been called, unlike
         `assert_called_with` and `assert_called_once_with` that only pass if
         the call is the most recent one."""
-        expected = self._call_matcher((args, kwargs))
+        expected = self._call_matcher(_Call((args, kwargs), two=True))
+        cause = expected if isinstance(expected, Exception) else None
         actual = [self._call_matcher(c) for c in self.call_args_list]
-        if expected not in actual:
-            cause = expected if isinstance(expected, Exception) else None
+        if cause or expected not in _AnyComparer(actual):
             expected_string = self._format_mock_call_signature(args, kwargs)
             raise AssertionError(
                 '%s call not found' % expected_string
@@ -1019,6 +1020,22 @@
         return f"\n{prefix}: {safe_repr(self.mock_calls)}."
 
 
+class _AnyComparer(list):
+    """A list which checks if it contains a call which may have an
+    argument of ANY, flipping the components of item and self from
+    their traditional locations so that ANY is guaranteed to be on
+    the left."""
+    def __contains__(self, item):
+        for _call in self:
+            if len(item) != len(_call):
+                continue
+            if all([
+                expected == actual
+                for expected, actual in zip(item, _call)
+            ]):
+                return True
+        return False
+
 
 def _try_iter(obj):
     if obj is None:
@@ -2171,9 +2188,9 @@
             msg = self._format_mock_failure_message(args, kwargs, action='await')
             return msg
 
-        expected = self._call_matcher((args, kwargs))
+        expected = self._call_matcher(_Call((args, kwargs), two=True))
         actual = self._call_matcher(self.await_args)
-        if expected != actual:
+        if actual != expected:
             cause = expected if isinstance(expected, Exception) else None
             raise AssertionError(_error_message()) from cause
 
@@ -2192,10 +2209,10 @@
         """
         Assert the mock has ever been awaited with the specified arguments.
         """
-        expected = self._call_matcher((args, kwargs))
+        expected = self._call_matcher(_Call((args, kwargs), two=True))
+        cause = expected if isinstance(expected, Exception) else None
         actual = [self._call_matcher(c) for c in self.await_args_list]
-        if expected not in actual:
-            cause = expected if isinstance(expected, Exception) else None
+        if cause or expected not in _AnyComparer(actual):
             expected_string = self._format_mock_call_signature(args, kwargs)
             raise AssertionError(
                 '%s await not found' % expected_string
diff --git a/Lib/unittest/test/testmock/testasync.py b/Lib/unittest/test/testmock/testasync.py
index 5502342..87f3cfc 100644
--- a/Lib/unittest/test/testmock/testasync.py
+++ b/Lib/unittest/test/testmock/testasync.py
@@ -2,8 +2,8 @@
 import inspect
 import unittest
 
-from unittest.mock import (call, AsyncMock, patch, MagicMock, create_autospec,
-                           _AwaitEvent)
+from unittest.mock import (ANY, call, AsyncMock, patch, MagicMock,
+                           create_autospec, _AwaitEvent)
 
 
 def tearDownModule():
@@ -192,6 +192,10 @@
         spec.assert_awaited_with(1, 2, c=3)
         spec.assert_awaited()
 
+        with self.assertRaises(AssertionError):
+            spec.assert_any_await(e=1)
+
+
     def test_patch_with_autospec(self):
 
         async def test_async():
@@ -607,6 +611,30 @@
         asyncio.run(self._runnable_test('SomethingElse'))
         self.mock.assert_has_awaits(calls)
 
+    def test_awaits_asserts_with_any(self):
+        class Foo:
+            def __eq__(self, other): pass
+
+        asyncio.run(self._runnable_test(Foo(), 1))
+
+        self.mock.assert_has_awaits([call(ANY, 1)])
+        self.mock.assert_awaited_with(ANY, 1)
+        self.mock.assert_any_await(ANY, 1)
+
+    def test_awaits_asserts_with_spec_and_any(self):
+        class Foo:
+            def __eq__(self, other): pass
+
+        mock_with_spec = AsyncMock(spec=Foo)
+
+        async def _custom_mock_runnable_test(*args):
+            await mock_with_spec(*args)
+
+        asyncio.run(_custom_mock_runnable_test(Foo(), 1))
+        mock_with_spec.assert_has_awaits([call(ANY, 1)])
+        mock_with_spec.assert_awaited_with(ANY, 1)
+        mock_with_spec.assert_any_await(ANY, 1)
+
     def test_assert_has_awaits_ordered(self):
         calls = [call('NormalFoo'), call('baz')]
         with self.assertRaises(AssertionError):
diff --git a/Lib/unittest/test/testmock/testhelpers.py b/Lib/unittest/test/testmock/testhelpers.py
index f3c7acb..9e7ec5d 100644
--- a/Lib/unittest/test/testmock/testhelpers.py
+++ b/Lib/unittest/test/testmock/testhelpers.py
@@ -64,7 +64,28 @@
             self.assertEqual(expected, mock.mock_calls)
             self.assertEqual(mock.mock_calls, expected)
 
+    def test_any_no_spec(self):
+        # This is a regression test for bpo-37555
+        class Foo:
+            def __eq__(self, other): pass
 
+        mock = Mock()
+        mock(Foo(), 1)
+        mock.assert_has_calls([call(ANY, 1)])
+        mock.assert_called_with(ANY, 1)
+        mock.assert_any_call(ANY, 1)
+
+    def test_any_and_spec_set(self):
+        # This is a regression test for bpo-37555
+        class Foo:
+            def __eq__(self, other): pass
+
+        mock = Mock(spec=Foo)
+
+        mock(Foo(), 1)
+        mock.assert_has_calls([call(ANY, 1)])
+        mock.assert_called_with(ANY, 1)
+        mock.assert_any_call(ANY, 1)
 
 class CallTest(unittest.TestCase):
 
diff --git a/Misc/ACKS b/Misc/ACKS
index f321131..71e61c3 100644
--- a/Misc/ACKS
+++ b/Misc/ACKS
@@ -503,6 +503,7 @@
 Segev Finer
 Jeffrey Finkelstein
 Russell Finn
+Neal Finne
 Dan Finnie
 Nils Fischbeck
 Frederik Fix
@@ -1714,6 +1715,7 @@
 Daniel Urban
 Michael Urman
 Hector Urtubia
+Elizabeth Uselton
 Lukas Vacek
 Ville Vainio
 Yann Vaginay
diff --git a/Misc/NEWS.d/next/Library/2019-07-19-20-13-48.bpo-37555.S5am28.rst b/Misc/NEWS.d/next/Library/2019-07-19-20-13-48.bpo-37555.S5am28.rst
new file mode 100644
index 0000000..16d1d62
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2019-07-19-20-13-48.bpo-37555.S5am28.rst
@@ -0,0 +1,2 @@
+Fix `NonCallableMock._call_matcher` returning tuple instead of `_Call` object
+when `self._spec_signature` exists. Patch by Elizabeth Uselton
\ No newline at end of file