diff --git a/src/anyio/_backends/_asyncio.py b/src/anyio/_backends/_asyncio.py
index 1157abd2..5a0aa936 100644
--- a/src/anyio/_backends/_asyncio.py
+++ b/src/anyio/_backends/_asyncio.py
@@ -677,32 +677,33 @@ def __init__(self, parent_id: int | None, cancel_scope: CancelScope | None):
         self.cancel_scope = cancel_scope
 
 
-class TaskStateStore(MutableMapping["Awaitable[Any] | asyncio.Task", TaskState]):
+class TaskStateStore(
+    MutableMapping["Coroutine[Any, Any, Any] | asyncio.Task", TaskState]
+):
     def __init__(self) -> None:
-        self._task_states = WeakKeyDictionary[
-            "asyncio.Task | Awaitable[Any]", TaskState
-        ]()
-        self._preliminary_task_states: dict[Awaitable[Any], TaskState] = {}
+        self._task_states = WeakKeyDictionary[asyncio.Task, TaskState]()
+        self._preliminary_task_states: dict[Coroutine[Any, Any, Any], TaskState] = {}
 
-    def __getitem__(self, key: Awaitable[Any] | asyncio.Task, /) -> TaskState:
+    def __getitem__(self, key: Coroutine[Any, Any, Any] | asyncio.Task, /) -> TaskState:
+        task = cast(asyncio.Task, key)
         try:
-            return self._task_states[key]
+            return self._task_states[task]
         except KeyError:
-            if coro := cast(asyncio.Task, key).get_coro():
+            if coro := task.get_coro():
                 if state := self._preliminary_task_states.get(coro):
                     return state
 
         raise KeyError(key)
 
     def __setitem__(
-        self, key: asyncio.Task | Awaitable[Any], value: TaskState, /
+        self, key: asyncio.Task | Coroutine[Any, Any, Any], value: TaskState, /
     ) -> None:
         if isinstance(key, Coroutine):
             self._preliminary_task_states[key] = value
         else:
             self._task_states[key] = value
 
-    def __delitem__(self, key: asyncio.Task | Awaitable[Any], /) -> None:
+    def __delitem__(self, key: asyncio.Task | Coroutine[Any, Any, Any], /) -> None:
         if isinstance(key, Coroutine):
             del self._preliminary_task_states[key]
         else:
@@ -711,7 +712,7 @@ def __delitem__(self, key: asyncio.Task | Awaitable[Any], /) -> None:
     def __len__(self) -> int:
         return len(self._task_states) + len(self._preliminary_task_states)
 
-    def __iter__(self) -> Iterator[Awaitable[Any] | asyncio.Task]:
+    def __iter__(self) -> Iterator[Coroutine[Any, Any, Any] | asyncio.Task]:
         yield from self._task_states
         yield from self._preliminary_task_states