| Somewhere you have a switch which turns the dome light on, off, or on | only when the doors are open.
Turns out that this was indeed the problem.
There's a switch that controls the brightness of the instrument lights, _and when turned all the way up it turns the dome light on_.
I've had this in previous cars, but didn't realize that the Explorer had it too -- we've never used it. Turn the switch off, lights go off! Funny how that works! And after mentioning it, oh yeah, my wife _did_ adjust the instrument brightness ...
Embarrassing to make such a silly mistake, but glad it's fixed, and it only cost me $10 for a new relay (which I'll just save for a spare, since they do burn out from time to time.)
Thanks for your help guys!