What are the steps needed to reproduce this symptom?
Are tabs involved?   Does it only happen with an inactive tab?

Have you tried using  TweakUI  with its maximal setting for preventing
focus stealing to see if your symptom will change?

FWIW  I have seen examples where prompt messages in background tabs
are not reshown when I try to see them but I can usually tell that they should
have appeared by the combination of the Task bar's icon flashing (e.g. due
to the TweakUI mod) and the read gleam on the tab of the target tab.
I have never seen an  iexplore.exe  task hung by such a scenario;
however, I must admit that I rarely minimize windows.

Hmm... a common example of the one that I am thinking of could fairly easily
be reproduced with the additional condition that the window be minimized.
E.g. taking advantage of the quirk that tabs opened by clicking on links
in OE messages are reused, we could open an IE window by:
1.    click on a link in an OE message.
2.    open another tab in that IE window.
3.    set Work Offline
4.    minimize that IE window
5.    switch back to the OE message (or a different one)
6.    click on a link in an OE message

Expectation would be a prompt message about the page not being available
offline, at least once I reopened that tab.

Results are:
1.    Flashing Task bar icon
2.    Red Task bar item (if multiple IE windows are currently open)
3.    Red background tab for the target window but no prompt (BUG)

Aha!  So that's how you could reproduce your symptom too.
E.g. if you had an OE message which contained a link to a page
which reliably would cause the prompt about mixed content
and you repeated the above steps (except for the Work Offline
command) you would have a case where a background tab
in a minimized window could be made to generate a prompt.
From my experience the worst symptom that I would expect
from that scenario is not seeing the prompt.   E.g. I would
expect to be able to switch to the target tab and then either
press F5 (if the Address bar had been updated--not likely)
or go back to the OE message and click on the link again,
which would allow the prompt to be shown.

So, what is an example of publicly accessible page
which can cause the prompt?   ; )

HTH

Robert Aldwinckle
---