Fix issue with running middleware context manager

Reported in encode#1678 (comment)

tests/middleware/test_base.py

@@ -1,4 +1,5 @@
 import contextvars
+from contextlib import AsyncExitStack
 
 import anyio
 import pytest
@@ -261,6 +262,69 @@ async def send(message):
     assert background_task_run.is_set()
 
 
+@pytest.mark.anyio
+async def test_run_context_manager_exit_even_if_client_disconnects():
+    # test for https://github.com/encode/starlette/issues/1678#issuecomment-1172916042
+    request_body_sent = False
+    response_complete = anyio.Event()
+    context_manager_exited = anyio.Event()
+
+    async def sleep_and_set():
+        # small delay to give BaseHTTPMiddleware a chance to cancel us
+        # this is required to make the test fail prior to fixing the issue
+        # so do not be surprised if you remove it and the test still passes
+        await anyio.sleep(0.1)
+        context_manager_exited.set()
+
+    class ContextManagerMiddleware:
+        def __init__(self, app):
+            self.app = app
+
+        async def __call__(self, scope: Scope, receive: Receive, send: Send):
+            async with AsyncExitStack() as stack:
+                stack.push_async_callback(sleep_and_set)
+                await self.app(scope, receive, send)
+
+    async def simple_endpoint(_):
+        return PlainTextResponse(background=BackgroundTask(sleep_and_set))
+
+    async def passthrough(request, call_next):
+        return await call_next(request)
+
+    app = Starlette(
+        middleware=[
+            Middleware(BaseHTTPMiddleware, dispatch=passthrough),
+            Middleware(ContextManagerMiddleware),
+        ],
+        routes=[Route("/", simple_endpoint)],
+    )
+
+    scope = {
+        "type": "http",
+        "version": "3",
+        "method": "GET",
+        "path": "/",
+    }
+
+    async def receive():
+        nonlocal request_body_sent
+        if not request_body_sent:
+            request_body_sent = True
+            return {"type": "http.request", "body": b"", "more_body": False}
+        # We simulate a client that disconnects immediately after receiving the response
+        await response_complete.wait()
+        return {"type": "http.disconnect"}
+
+    async def send(message):
+        if message["type"] == "http.response.body":
+            if not message.get("more_body", False):
+                response_complete.set()
+
+    await app(scope, receive, send)
+
+    assert context_manager_exited.is_set()
+
+
 def test_app_receives_http_disconnect_while_sending_if_discarded(test_client_factory):
     class DiscardingMiddleware(BaseHTTPMiddleware):
         async def dispatch(self, request, call_next):