diff --git a/mypy/messages.py b/mypy/messages.py
index b63310825f7d7..8e614f02277a9 100644
--- a/mypy/messages.py
+++ b/mypy/messages.py
@@ -2855,7 +2855,29 @@ def format_type_distinctly(*types: Type, options: Options, bare: bool = False) -
     quoting them (such as prepending * or **) should use this.
     """
     overlapping = find_type_overlaps(*types)
-    for verbosity in range(2):
+
+    def format_single(arg: Type) -> str:
+        return format_type_inner(arg, verbosity=0, options=options, fullnames=overlapping)
+
+    min_verbosity = 0
+    # Prevent emitting weird errors like:
+    # ... has incompatible type "Callable[[int], Child]"; expected "Callable[[int], Parent]"
+    if len(types) == 2:
+        left, right = types
+        left = get_proper_type(left)
+        right = get_proper_type(right)
+        # If the right type has named arguments, they may be the reason for incompatibility.
+        # This excludes cases when right is Callable[[Something], None] without named args,
+        # because that's usually the right thing to do.
+        if (
+            isinstance(left, CallableType)
+            and isinstance(right, CallableType)
+            and any(right.arg_names)
+            and is_subtype(left, right, ignore_pos_arg_names=True)
+        ):
+            min_verbosity = 1
+
+    for verbosity in range(min_verbosity, 2):
         strs = [
             format_type_inner(type, verbosity=verbosity, options=options, fullnames=overlapping)
             for type in types
diff --git a/test-data/unit/check-functions.test b/test-data/unit/check-functions.test
index 18425efb9cb08..58973307a1ae7 100644
--- a/test-data/unit/check-functions.test
+++ b/test-data/unit/check-functions.test
@@ -3472,3 +3472,51 @@ class Qux(Bar):
     def baz(self, x) -> None:
         pass
 [builtins fixtures/tuple.pyi]
+
+[case testDistinctFormatting]
+from typing import Awaitable, Callable, ParamSpec
+
+P = ParamSpec("P")
+
+class A: pass
+class B(A): pass
+
+def decorator(f: Callable[P, None]) -> Callable[[Callable[P, A]], None]:
+    return lambda _: None
+
+def key(x: int) -> None: ...
+def fn_b(b: int) -> B: ...
+
+decorator(key)(fn_b)  # E: Argument 1 has incompatible type "Callable[[Arg(int, 'b')], B]"; expected "Callable[[Arg(int, 'x')], A]"
+
+def decorator2(f: Callable[P, None]) -> Callable[
+    [Callable[P, Awaitable[None]]],
+    Callable[P, Awaitable[None]],
+]:
+    return lambda f: f
+
+def key2(x: int) -> None:
+    ...
+
+@decorator2(key2)  # E: Argument 1 has incompatible type "Callable[[Arg(int, 'y')], Coroutine[Any, Any, None]]"; expected "Callable[[Arg(int, 'x')], Awaitable[None]]"
+async def foo2(y: int) -> None:
+    ...
+
+class Parent:
+    def method_without(self) -> "Parent": ...
+    def method_with(self, param: str) -> "Parent": ...
+
+class Child(Parent):
+    method_without: Callable[["Child"], "Child"]
+    method_with: Callable[["Child", str], "Child"]  # E: Incompatible types in assignment (expression has type "Callable[[str], Child]", base class "Parent" defined the type as "Callable[[Arg(str, 'param')], Parent]")
+[builtins fixtures/tuple.pyi]
+
+[case testDistinctFormattingUnion]
+from typing import Callable, Union
+from mypy_extensions import Arg
+
+def f(x: Callable[[Arg(int, 'x')], None]) -> None: pass
+
+y: Callable[[Union[int, str]], None]
+f(y)  # E: Argument 1 to "f" has incompatible type "Callable[[Union[int, str]], None]"; expected "Callable[[Arg(int, 'x')], None]"
+[builtins fixtures/tuple.pyi]