diff --git a/docs/versionhistory.rst b/docs/versionhistory.rst
index 81f656aec..b027e1c67 100644
--- a/docs/versionhistory.rst
+++ b/docs/versionhistory.rst
@@ -75,6 +75,9 @@ This library adheres to `Semantic Versioning 2.0 <http://semver.org/>`_.
 - Fixed an asyncio worker thread race that could raise ``RuntimeError`` when the event
   loop closed between checking its state and scheduling the worker result
   (`#1265 <https://github.com/agronholm/anyio/issues/1265>`_; PR by @hansu650)
+- Fixed concurrent ``aclose_forcefully()`` calls on asyncio socket streams returning
+  before the underlying socket was closed
+  (`#1273 <https://github.com/agronholm/anyio/issues/1273>`_; PR by @hansu650)
 - Fixed ``CapacityLimiter`` on the asyncio backend over-granting tokens when
   ``total_tokens`` was raised while the limiter was over-subscribed
   (`#1223 <https://github.com/agronholm/anyio/pull/1223>`_; PR by @zelinewang)
diff --git a/src/anyio/_backends/_asyncio.py b/src/anyio/_backends/_asyncio.py
index 35b0cf371..b3498f6e4 100644
--- a/src/anyio/_backends/_asyncio.py
+++ b/src/anyio/_backends/_asyncio.py
@@ -1397,15 +1397,8 @@ async def send_eof(self) -> None:
 
     async def aclose(self) -> None:
         self._closed = True
-        if not self._transport.is_closing():
-            try:
-                self._transport.write_eof()
-            except OSError:
-                pass
-
-            self._transport.close()
-            await sleep(0)
-            self._transport.abort()
+        self._transport.abort()
+        await sleep(0)
 
 
 class _RawSocketMixin:
@@ -1689,7 +1682,8 @@ async def aclose(self) -> None:
         if not self._transport.is_closing():
             self._transport.close()
 
-        await self._protocol.closed_event.wait()
+        with CancelScope(shield=True):
+            await self._protocol.closed_event.wait()
 
     async def receive(self) -> tuple[bytes, IPSockAddrType]:
         with self._receive_guard:
@@ -1739,7 +1733,8 @@ async def aclose(self) -> None:
         if not self._transport.is_closing():
             self._transport.close()
 
-        await self._protocol.closed_event.wait()
+        with CancelScope(shield=True):
+            await self._protocol.closed_event.wait()
 
     async def receive(self) -> bytes:
         with self._receive_guard:
diff --git a/tests/test_sockets.py b/tests/test_sockets.py
index cc661b79a..a91700875 100644
--- a/tests/test_sockets.py
+++ b/tests/test_sockets.py
@@ -37,12 +37,14 @@
 from anyio import (
     BrokenResourceError,
     BusyResourceError,
+    CancelScope,
     ClosedResourceError,
     EndOfStream,
     Event,
     TCPConnectable,
     TypedAttributeLookupError,
     UNIXConnectable,
+    aclose_forcefully,
     as_connectable,
     connect_tcp,
     connect_unix,
@@ -54,6 +56,7 @@
     create_unix_datagram_socket,
     create_unix_listener,
     fail_after,
+    get_cancelled_exc_class,
     getaddrinfo,
     getnameinfo,
     move_on_after,
@@ -508,6 +511,40 @@ async def interrupt() -> None:
                 with pytest.raises(ClosedResourceError):
                     await stream.receive()
 
+    async def test_concurrent_aclose_forcefully_waits_for_socket_close(
+        self, server_addr: tuple[str, int]
+    ) -> None:
+        stream = await connect_tcp(*server_addr)
+        raw_socket = stream.extra(SocketAttribute.raw_socket)
+        file_descriptors: list[int] = []
+
+        async def close_stream() -> None:
+            await aclose_forcefully(stream)
+            file_descriptors.append(raw_socket.fileno())
+
+        async with create_task_group() as task_group:
+            task_group.start_soon(close_stream)
+            task_group.start_soon(close_stream)
+
+        assert file_descriptors == [-1, -1]
+
+    @pytest.mark.parametrize("anyio_backend", asyncio_params)
+    async def test_aclose_propagates_cancellation(
+        self, server_addr: tuple[str, int]
+    ) -> None:
+        stream = await connect_tcp(*server_addr)
+        cancelled_exc: BaseException | None = None
+
+        with CancelScope() as scope:
+            scope.cancel()
+            try:
+                await stream.aclose()
+            except get_cancelled_exc_class() as exc:
+                cancelled_exc = exc
+                raise
+
+        assert cancelled_exc is not None
+
     async def test_receive_after_close(self, server_addr: tuple[str, int]) -> None:
         stream = await connect_tcp(*server_addr)
         await stream.aclose()
@@ -1764,6 +1801,17 @@ async def handle(stream: SocketStream) -> None:
 @pytest.mark.network
 @pytest.mark.usefixtures("check_asyncio_bug")
 class TestUDPSocket:
+    async def test_aclose_forcefully_waits_for_fd_release(
+        self, family: AnyIPAddressFamily
+    ) -> None:
+        host = "127.0.0.1" if family == socket.AF_INET else "::1"
+        udp = await create_udp_socket(family=family, local_host=host)
+        raw_socket = udp.extra(SocketAttribute.raw_socket)
+
+        await aclose_forcefully(udp)
+
+        assert raw_socket.fileno() == -1
+
     async def test_aclose_waits_for_fd_release(
         self, family: AnyIPAddressFamily, free_udp_port: int
     ) -> None:
@@ -1951,6 +1999,22 @@ async def test_from_socket_wrong_socket_type(
 @pytest.mark.network
 @pytest.mark.usefixtures("check_asyncio_bug")
 class TestConnectedUDPSocket:
+    async def test_aclose_forcefully_waits_for_fd_release(
+        self, family: AnyIPAddressFamily
+    ) -> None:
+        host = "127.0.0.1" if family == socket.AF_INET else "::1"
+        peer = socket.socket(family, socket.SOCK_DGRAM)
+        peer.bind((host, 0))
+        try:
+            udp = await create_connected_udp_socket(*peer.getsockname()[:2])
+            raw_socket = udp.extra(SocketAttribute.raw_socket)
+
+            await aclose_forcefully(udp)
+
+            assert raw_socket.fileno() == -1
+        finally:
+            peer.close()
+
     async def test_aclose_waits_for_fd_release(
         self, family: AnyIPAddressFamily, free_udp_port: int
     ) -> None: