I am mildly intrigued by this comment in icomplete-post-command-hook:

  (defun icomplete-post-command-hook ()
    (let ((non-essential t)) ;E.g. don't prompt for password!
      (icomplete-exhibit)))

As can be seen in the backtraces, prompts are definitely happening while
calling icomplete-exhibit; I have no idea whether this is suspicious, or
if that comment is a red herring…

My hunch is that icomplete sets up *something* that is not cleared and
keeps being triggered when the second prompt shows up; I haven't been
able to pinpoint what yet.

If any completion wizard out here has advice on how to debug this
further, I'm all ears!