I did look at Console logs, and did not see anything obvious to me. So, I am asking if there is some debugging that can be enabled so that there is some message in the logs to tell me what it is waiting for. Since the screen is already blank, and there is the rotating circle in the lower middle of the screen I would guess that Finder is already shut down, and my User has logged out? Since the "bong" has not yet been heard... I suppose that the Darwin kernel is still running and waiting for something to finish. (Just guesswork on my part.)
I've left the machine up four days, without quitting my startups, and have rebooted normally. The condition seems to happen only when my iMac has been up a long time. Perhaps it is just one of those things that I have to live with, but thought I would at least see anyone more experience in MacOS had any ideas on how the culprit could be identified. This issue goes back to El Capitan days, and seems to happen less frequently with Mojave, so there may be something with my iMac with external display and disks. (Seems, because I may just reboot less often with Mojave.)
I do have two external USB disks, one with two partitions and the other with three including the eDrive from TTP, so perhaps one of the disks cannot be 'ejected'. If a file is left open, I don't think the screen would be blank implying that Finder/User are gone... (my guess again)