diff --git a/ipykernel/kernelbase.py b/ipykernel/kernelbase.py
index 6465751c..263ff209 100644
--- a/ipykernel/kernelbase.py
+++ b/ipykernel/kernelbase.py
@@ -428,6 +428,8 @@ async def shell_main(self, subshell_id: str | None):
                 await to_thread.run_sync(self.shell_stop.wait)
                 tg.cancel_scope.cancel()
 
+        await socket.stop()
+
     async def process_shell(self, socket=None):
         # socket=None is valid if kernel subshells are not supported.
         try:
diff --git a/ipykernel/subshell.py b/ipykernel/subshell.py
index 180e9ecb..34aa6efe 100644
--- a/ipykernel/subshell.py
+++ b/ipykernel/subshell.py
@@ -32,11 +32,4 @@ async def create_pair_socket(
         self._pair_socket = zmq_anyio.Socket(context, zmq.PAIR)
         self._pair_socket.connect(address)
         self.start_soon(self._pair_socket.start)
-
-    def run(self) -> None:
-        try:
-            super().run()
-        finally:
-            if self._pair_socket is not None:
-                self._pair_socket.close()
-                self._pair_socket = None
+        self.add_teardown_callback(self._pair_socket.stop)
diff --git a/ipykernel/subshell_manager.py b/ipykernel/subshell_manager.py
index f4f92d2d..4068f27e 100644
--- a/ipykernel/subshell_manager.py
+++ b/ipykernel/subshell_manager.py
@@ -116,11 +116,13 @@ def close(self) -> None:
     async def get_control_other_socket(self, thread: BaseThread) -> zmq_anyio.Socket:
         if not self._control_other_socket.started.is_set():
             await thread.task_group.start(self._control_other_socket.start)
+            thread.add_teardown_callback(self._control_other_socket.stop)
         return self._control_other_socket
 
     async def get_control_shell_channel_socket(self, thread: BaseThread) -> zmq_anyio.Socket:
         if not self._control_shell_channel_socket.started.is_set():
             await thread.task_group.start(self._control_shell_channel_socket.start)
+            thread.add_teardown_callback(self._control_shell_channel_socket.stop)
         return self._control_shell_channel_socket
 
     def get_other_socket(self, subshell_id: str | None) -> zmq_anyio.Socket:
@@ -281,6 +283,8 @@ async def _listen_for_subshell_reply(
                 # Subshell no longer exists so exit gracefully
                 return
             raise
+        finally:
+            await shell_channel_socket.stop()
 
     async def _process_control_request(
         self, request: dict[str, t.Any], subshell_task: t.Any
diff --git a/ipykernel/thread.py b/ipykernel/thread.py
index a66cb2a4..185fd8d8 100644
--- a/ipykernel/thread.py
+++ b/ipykernel/thread.py
@@ -3,6 +3,7 @@
 from __future__ import annotations
 
 from collections.abc import Awaitable
+from inspect import isawaitable
 from queue import Queue
 from threading import Event, Thread
 from typing import Any, Callable
@@ -26,6 +27,7 @@ def __init__(self, **kwargs):
         self.is_pydev_daemon_thread = True
         self._tasks: Queue[tuple[str, Callable[[], Awaitable[Any]]] | None] = Queue()
         self._result: Queue[Any] = Queue()
+        self._teardown_callbacks: list[Callable[[], Any] | Callable[[], Awaitable[Any]]] = []
         self._exception: Exception | None = None
 
     @property
@@ -47,6 +49,9 @@ def run_sync(self, func: Callable[..., Any]) -> Any:
         self._tasks.put(("run_sync", func))
         return self._result.get()
 
+    def add_teardown_callback(self, func: Callable[[], Any] | Callable[[], Awaitable[Any]]) -> None:
+        self._teardown_callbacks.append(func)
+
     def run(self) -> None:
         """Run the thread."""
         try:
@@ -55,24 +60,37 @@ def run(self) -> None:
             self._exception = exc
 
     async def _main(self) -> None:
-        async with create_task_group() as tg:
-            self._task_group = tg
-            self.started.set()
-            while True:
-                task = await to_thread.run_sync(self._tasks.get)
-                if task is None:
-                    break
-                func, arg = task
-                if func == "start_soon":
-                    tg.start_soon(arg)
-                elif func == "run_async":
-                    res = await arg
-                    self._result.put(res)
-                else:  # func == "run_sync"
-                    res = arg()
-                    self._result.put(res)
-
-            tg.cancel_scope.cancel()
+        try:
+            async with create_task_group() as tg:
+                self._task_group = tg
+                self.started.set()
+                while True:
+                    task = await to_thread.run_sync(self._tasks.get)
+                    if task is None:
+                        break
+                    func, arg = task
+                    if func == "start_soon":
+                        tg.start_soon(arg)
+                    elif func == "run_async":
+                        res = await arg
+                        self._result.put(res)
+                    else:  # func == "run_sync"
+                        res = arg()
+                        self._result.put(res)
+
+                tg.cancel_scope.cancel()
+        finally:
+            exception = None
+            for teardown_callback in self._teardown_callbacks[::-1]:
+                try:
+                    res = teardown_callback()
+                    if isawaitable(res):
+                        await res
+                except Exception as exc:
+                    if exception is None:
+                        exception = exc
+            if exception is not None:
+                raise exception
 
     def stop(self) -> None:
         """Stop the thread.
diff --git a/pyproject.toml b/pyproject.toml
index 1853544f..4bd8721c 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -33,7 +33,7 @@ dependencies = [
     "psutil>=5.7",
     "packaging>=22",
     "anyio>=4.8.0,<5.0.0",
-    "zmq-anyio >=0.3.6",
+    "zmq-anyio >=0.3.9",
 ]
 
 [project.urls]