>> >From emacs -Q:
 >>
 >> M-x set-variable RET pop-up-frames RET t RET
 >> M-x global- TAB
 >>
 >> The *Completions* buffer is opened in a new frame, but the cursor is in
 >> the frame too, so the user has to switch back to the minibuffer to
 >> continue typing.
 >
 > This problem is still present in Emacs 29.  I guess the general solution
 > here would be for completion to ensure that it's gotten focus back again
 > after displaying the *Completions* buffer?

Maybe Stefan Monnier can tell us what he does in such case.

I can offer the attached, largely untested patch.  A possible
customization would then be

(customize-set-variable
  'display-buffer-alist
  '(("*Completions*"
     (display-buffer-reuse-window
      display-buffer-pop-up-frame)
     (reusable-frames . t)
     (redirect-frame-focus . t))))

martin