Now try doing the same thing but using C-x 5 o to switch the frames will
cause the popup to work just fine.  Does C-x 5 o some thing special
compared to Alt-tab?

      

It gives focus to an Emacs window. The same is seen if you switch focus
by clicking on the title bar vs in the text area.
    

But C-x 5 2 seems to do this too.