The spec currently says

Authors SHOULD ensure that elements with the role tooltip are referenced through the use of aria-describedby before or at the time the tooltip is displayed.

It seems like aria-details might be a good alternative for authors and users (especially when done automatically via w3c/html-aam#545).

From a conversation with @scottaohara, the combination might also be useful, e.g.,

<a href=... aria-describedby=t aria-details=t>foo</a>

<div role=tooltip id=t aria-labelledby=h>
  <h3 id=h>whatever</h3>
  <p>all the slop here</p>
</div>

The describedBy could provide a brief summary (since the accDescription computation should stop once it gets the name of the tooltip) and the aria-details could be the signal that there's more content and an easy way to fully access it. (This example might also make it a candidate for #2215)

Maybe we can even resolve #979 with this new information.