diff --git a/stdlib/collections/__init__.pyi b/stdlib/collections/__init__.pyi
index 08bc521d0040..e534c13c026b 100644
--- a/stdlib/collections/__init__.pyi
+++ b/stdlib/collections/__init__.pyi
@@ -266,15 +266,16 @@ class Counter(dict[_T, int], Generic[_T]):
     def subtract(self, __mapping: Mapping[_T, int]) -> None: ...
     @overload
     def subtract(self, __iterable: Iterable[_T]) -> None: ...
-    # The Iterable[Tuple[...]] argument type is not actually desirable
-    # (the tuples will be added as keys, breaking type safety) but
-    # it's included so that the signature is compatible with
-    # Dict.update. Not sure if we should use '# type: ignore' instead
-    # and omit the type from the union.
+    # Unlike dict.update(), use Mapping instead of SupportsKeysAndGetItem for the first overload
+    # (source code does an `isinstance(other, Mapping)` check)
+    #
+    # The second overload is also deliberately different to dict.update()
+    # (if it were `Iterable[_T] | Iterable[tuple[_T, int]]`,
+    # the tuples would be added as keys, breaking type safety)
+    @overload  # type: ignore[override]
+    def update(self, __m: Mapping[_T, int], **kwargs: int) -> None: ...
     @overload
-    def update(self, __m: SupportsKeysAndGetItem[_T, int], **kwargs: int) -> None: ...
-    @overload
-    def update(self, __m: Iterable[_T] | Iterable[tuple[_T, int]], **kwargs: int) -> None: ...
+    def update(self, __m: Iterable[_T], **kwargs: int) -> None: ...
     @overload
     def update(self, __m: None = ..., **kwargs: int) -> None: ...
     def __missing__(self, key: _T) -> int: ...
@@ -283,22 +284,23 @@ class Counter(dict[_T, int], Generic[_T]):
         def __eq__(self, other: object) -> bool: ...
         def __ne__(self, other: object) -> bool: ...
 
-    def __add__(self, other: Counter[_T]) -> Counter[_T]: ...
+    def __add__(self, other: Counter[_S]) -> Counter[_T | _S]: ...
     def __sub__(self, other: Counter[_T]) -> Counter[_T]: ...
     def __and__(self, other: Counter[_T]) -> Counter[_T]: ...
-    def __or__(self, other: Counter[_T]) -> Counter[_T]: ...  # type: ignore[override]
+    def __or__(self, other: Counter[_S]) -> Counter[_T | _S]: ...  # type: ignore[override]
     def __pos__(self) -> Counter[_T]: ...
     def __neg__(self) -> Counter[_T]: ...
-    def __iadd__(self: Self, other: Counter[_T]) -> Self: ...
+    # several type: ignores because __iadd__ is supposedly incompatible with __add__, etc.
+    def __iadd__(self: Self, other: Counter[_T]) -> Self: ...  # type: ignore[misc]
     def __isub__(self: Self, other: Counter[_T]) -> Self: ...
     def __iand__(self: Self, other: Counter[_T]) -> Self: ...
-    def __ior__(self: Self, other: Counter[_T]) -> Self: ...  # type: ignore[override]
+    def __ior__(self: Self, other: Counter[_T]) -> Self: ...  # type: ignore[override,misc]
     if sys.version_info >= (3, 10):
         def total(self) -> int: ...
-        def __le__(self, other: Counter[object]) -> bool: ...
-        def __lt__(self, other: Counter[object]) -> bool: ...
-        def __ge__(self, other: Counter[object]) -> bool: ...
-        def __gt__(self, other: Counter[object]) -> bool: ...
+        def __le__(self, other: Counter[Any]) -> bool: ...
+        def __lt__(self, other: Counter[Any]) -> bool: ...
+        def __ge__(self, other: Counter[Any]) -> bool: ...
+        def __gt__(self, other: Counter[Any]) -> bool: ...
 
 @final
 class _OrderedDictKeysView(dict_keys[_KT_co, _VT_co], Reversible[_KT_co]):  # type: ignore[misc]