Have you tried using Spectrum Web Components yet to build your plugin UI?

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Is there something extra you are doing to get this to work? e.g. a web component wrapper? It doesn’t do anything for me.

No, there is nothing extra. As @JasonM mentioned such things as slot="trigger" are important.

I can’t for the life of me get sp-overlay to work - I’m also intrigued to where it’s documented as I can’t see it here (or sp-popover for that matter).

Yes, this is not documented. I’m not sure where I got the idea for the code. I think @kerrishotts posted some snippets here in the forum.