@LeaVerou raised the idea of adding a combinator that allows an author to select an element that is the target of an interest invoker. So:

<button id=invoker interestfor=target>Interest invoker</button>
<div popover id=target>Target of interest invoker</div>

<style>
  #invoker --> #target {
    /* Styles for the target popover that only apply when
      #foo points to #target, as above. */
  }
</style>

Here, I made up the combinator as --> just as a placeholder. Though it's not bad, maybe?

If this combinator is a good idea, it might alleviate some of the questions in #12154 (comment), e.g. by obviating the need for two separate pseudo classes for the invoker and the target. If there was simply a pseudo class on the invoker, then "target with interest" could be selected with :interest-invoker --> [popover].