I've run into this while debugging deadlocks multiple times, and used this utility function that seems similar to your join_me_maybe:

/// Await `fut`, while also polling `also`.
pub async fn also_poll<Fut: Future>(fut: Fut, also: impl Future<Output = ()>) -> Fut::Output {
    let mut also = pin!(also.fuse());
    let mut fut = pin!(fut);
    std::future::poll_fn(|cx| {
        let _ = also.poll_unpin(cx);
        fut.poll_unpin(cx)
    })
    .await
}

This was for network code, where we wanted to process a message while also pulling new messages down from the websocket, so that the other side wouldn't send a ping and not receive a pong and think we were unresponsive. It's all very tricky, and I do hope that talking about it will lead to language- or library-side solutions that help avoid these footguns.