diff --git a/mypy/checker.py b/mypy/checker.py
index fc9733117a0a..f9e9b9c0e95c 100644
--- a/mypy/checker.py
+++ b/mypy/checker.py
@@ -2234,6 +2234,9 @@ def check_method_override_for_base_with_name(
                 # Will always fail to typecheck below, since we know the node is a method
                 original_type = NoneType()
 
+        if isinstance(original_node, Var) and original_node.allow_incompatible_override:
+            return False
+
         always_allow_covariant = False
         if is_settable_property(defn) and (
             is_settable_property(original_node) or isinstance(original_node, Var)
@@ -3435,18 +3438,30 @@ def check_compatibility_all_supers(self, lvalue: RefExpr, rvalue: Expression) ->
                 return
 
             for base in lvalue_node.info.mro[1:]:
-                # The type of "__slots__" and some other attributes usually doesn't need to
-                # be compatible with a base class. We'll still check the type of "__slots__"
-                # against "object" as an exception.
-                if lvalue_node.allow_incompatible_override and not (
-                    lvalue_node.name == "__slots__" and base.fullname == "builtins.object"
+                if (
+                    lvalue_node.name == "__hash__"
+                    and base.fullname == "builtins.object"
+                    and isinstance(get_proper_type(lvalue_type), NoneType)
                 ):
+                    # allow `__hash__ = None` if the overridden `__hash__` comes from object
+                    # This isn't LSP-compliant, but too common in real code.
                     continue
 
                 if is_private(lvalue_node.name):
                     continue
 
                 base_type, base_node = self.node_type_from_base(lvalue_node.name, base, lvalue)
+                # The type of "__slots__" and some other attributes usually doesn't need to
+                # be compatible with a base class. We'll still check the type of "__slots__"
+                # against "object" as an exception.
+                if (
+                    isinstance(base_node, Var)
+                    and base_node.allow_incompatible_override
+                    and not (
+                        lvalue_node.name == "__slots__" and base.fullname == "builtins.object"
+                    )
+                ):
+                    continue
                 custom_setter = is_custom_settable_property(base_node)
                 if isinstance(base_type, PartialType):
                     base_type = None
diff --git a/mypy/semanal.py b/mypy/semanal.py
index d70abe911fea..9f678d6543b9 100644
--- a/mypy/semanal.py
+++ b/mypy/semanal.py
@@ -2018,6 +2018,27 @@ def analyze_class_body_common(self, defn: ClassDef) -> None:
             self.setup_self_type()
         defn.defs.accept(self)
         self.apply_class_plugin_hooks(defn)
+
+        if (
+            "__eq__" in defn.info.names
+            and "__hash__" not in defn.info.names
+            and not defn.info.is_protocol
+        ):
+            # If a class defines `__eq__` without `__hash__`, it's no longer hashable.
+            # Excludes Protocol from consideration as we don't want to enforce unhashability
+            # of their instances.
+            hash_none = Var("__hash__", NoneType())
+            hash_none.info = defn.info
+            hash_none.set_line(defn)
+            hash_none.is_classvar = True
+            # Making a class hashable is allowed even if its parents weren't.
+            # The only possible consequence of this LSP violation would be
+            # `assert child.__hash__ is None` no longer passing, probably nobody
+            # cares about that - it's impossible to statically restrict to non-hashable
+            # anyway.
+            hash_none.allow_incompatible_override = True
+            self.add_symbol("__hash__", hash_none, defn)
+
         self.leave_class()
 
     def analyze_typeddict_classdef(self, defn: ClassDef) -> bool:
@@ -3237,6 +3258,22 @@ def visit_assignment_stmt(self, s: AssignmentStmt) -> None:
         self.process__all__(s)
         self.process__deletable__(s)
         self.process__slots__(s)
+        self.process__hash__(s)
+
+    def process__hash__(self, s: AssignmentStmt) -> None:
+        # Allow overriding `__hash__ = None` in subclasses.
+        if (
+            isinstance(self.type, TypeInfo)
+            and len(s.lvalues) == 1
+            and isinstance(s.lvalues[0], NameExpr)
+            and s.lvalues[0].name == "__hash__"
+            and s.lvalues[0].kind == MDEF
+            and isinstance(s.rvalue, NameExpr)
+            and s.rvalue.name == "None"
+        ):
+            var = s.lvalues[0].node
+            if isinstance(var, Var):
+                var.allow_incompatible_override = True
 
     def analyze_identity_global_assignment(self, s: AssignmentStmt) -> bool:
         """Special case 'X = X' in global scope.
diff --git a/mypyc/test-data/fixtures/ir.py b/mypyc/test-data/fixtures/ir.py
index 1b92590a5fd4..5a5c65900bf4 100644
--- a/mypyc/test-data/fixtures/ir.py
+++ b/mypyc/test-data/fixtures/ir.py
@@ -90,6 +90,7 @@ def __init__(self, x: object) -> None: pass
     def __add__(self, x: str) -> str: pass
     def __mul__(self, x: int) -> str: pass
     def __rmul__(self, x: int) -> str: pass
+    def __hash__(self) -> int: pass
     def __eq__(self, x: object) -> bool: pass
     def __ne__(self, x: object) -> bool: pass
     def __lt__(self, x: str) -> bool: ...
diff --git a/test-data/unit/check-classes.test b/test-data/unit/check-classes.test
index f4bbaf41dc47..74a17f157b6f 100644
--- a/test-data/unit/check-classes.test
+++ b/test-data/unit/check-classes.test
@@ -116,11 +116,7 @@ class Base:
     __hash__: None = None
 
 class Derived(Base):
-    def __hash__(self) -> int:  # E: Signature of "__hash__" incompatible with supertype "Base" \
-                                # N:      Superclass: \
-                                # N:          None \
-                                # N:      Subclass: \
-                                # N:          def __hash__(self) -> int
+    def __hash__(self) -> int:
         pass
 
 # Correct:
@@ -147,6 +143,63 @@ class Base:
 class Derived(Base):
     __hash__ = 1  # E: Incompatible types in assignment (expression has type "int", base class "Base" defined the type as "Callable[[], int]")
 
+[case testEqWithoutHash]
+class A:
+    def __eq__(self, other) -> bool: ...
+
+reveal_type(A.__hash__)  # N: Revealed type is "None"
+[builtins fixtures/primitives.pyi]
+
+[case testHashNoneOverride]
+class A:
+    __hash__ = None
+
+class B(A):
+    def __hash__(self) -> int:
+        return 0
+
+class C(A):
+    __hash__ = object.__hash__
+
+class D(A):
+    def __hash__(self, x: int) -> str:  # E: Signature of "__hash__" incompatible with supertype "builtins.object" \
+                                        # N:      Superclass: \
+                                        # N:          def __hash__(self) -> int \
+                                        # N:      Subclass: \
+                                        # N:          def __hash__(self, x: int) -> str
+        return ''
+
+def bad_hash(x: E, y: str) -> str:
+    return y
+
+class E(A):
+    __hash__ = bad_hash  # E: Incompatible types in assignment (expression has type "Callable[[str], str]", base class "object" defined the type as "Callable[[], int]")
+
+
+
+
+[builtins fixtures/primitives.pyi]
+
+[case testHashNoneBadOverride]
+class A:
+    def __hash__(self) -> int: return 0
+
+class B(A):
+    __hash__ = None  # E: Incompatible types in assignment (expression has type "None", base class "A" defined the type as "Callable[[], int]")
+[builtins fixtures/primitives.pyi]
+
+[case testHashOverrideInMultipleInheritance]
+class Foo:
+    def __eq__(self, other: "Foo") -> bool: ...
+
+class Bar:
+    def __hash__(self) -> int: ...
+
+class BadChild(Foo, Bar): ...  # E: Definition of "__hash__" in base class "Foo" is incompatible with definition in base class "Bar"
+class GoodChild(Bar, Foo): ...
+[builtins fixtures/tuple.pyi]
+
+
 [case testOverridePartialAttributeWithMethod]
 # This was crashing: https://github.com/python/mypy/issues/11686.
 class Base:
diff --git a/test-data/unit/fine-grained-inspect.test b/test-data/unit/fine-grained-inspect.test
index 5caa1a94387b..b40669ae82a3 100644
--- a/test-data/unit/fine-grained-inspect.test
+++ b/test-data/unit/fine-grained-inspect.test
@@ -122,10 +122,10 @@ None
 [builtins fixtures/args.pyi]
 [out]
 ==
-{"C": ["x"], "object": ["__eq__", "__init__", "__ne__"]}
+{"C": ["x"], "object": ["__eq__", "__hash__", "__init__", "__ne__"]}
 {"Iterable": ["__iter__"]}
 NameExpr -> {}
-NameExpr -> {"object": ["__eq__", "__init__", "__ne__"]}
+NameExpr -> {"object": ["__eq__", "__hash__", "__init__", "__ne__"]}
 
 [case testInspectTypeVarBoundAttrs]
 # inspect2: --show=attrs tmp/foo.py:8:13
diff --git a/test-data/unit/fixtures/primitives.pyi b/test-data/unit/fixtures/primitives.pyi
index 2f8623c79b9f..22e1b2bb8302 100644
--- a/test-data/unit/fixtures/primitives.pyi
+++ b/test-data/unit/fixtures/primitives.pyi
@@ -10,6 +10,7 @@ class object:
     def __str__(self) -> str: pass
     def __eq__(self, other: object) -> bool: pass
     def __ne__(self, other: object) -> bool: pass
+    def __hash__(self) -> int: pass
 
 class type:
     def __init__(self, x: object) -> None: pass