diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 77ba66925..bf337b8e9 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -34,6 +34,10 @@ async def coro() -> None:
             try:
                 message = await recv_stream.receive()
             except anyio.EndOfStream:
+                # HACK: give anyio a chance to surface any app exception first,
+                # in order to avoid an `anyio.ExceptionGroup`.
+                # See #1255.
+                await anyio.lowlevel.checkpoint()
                 raise RuntimeError("No response returned.")
 
             assert message["type"] == "http.response.start"
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index 8a8df4ea6..c6bfd49fc 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -25,7 +25,7 @@ def homepage(request):
 
 @app.route("/exc")
 def exc(request):
-    raise Exception()
+    raise Exception("Exc")
 
 
 @app.route("/no-response")
@@ -52,8 +52,9 @@ def test_custom_middleware(test_client_factory):
     response = client.get("/")
     assert response.headers["Custom-Header"] == "Example"
 
-    with pytest.raises(Exception):
+    with pytest.raises(Exception) as ctx:
         response = client.get("/exc")
+    assert str(ctx.value) == "Exc"
 
     with pytest.raises(RuntimeError):
         response = client.get("/no-response")