I'm just posting this in case anyone else experiences the same problem.
I have my viewfinder set to Auto, meaning that without my eye to it, the rear monitor works. When I put my eye up the viewfinder, it turns on and the monitor turns off. This is normal, I suspect, for most of us.
However, yesterday I could not get the monitor to go on when I took my eye from the viewfinder. It would work if I changed the menu to Manual, but otherwise it stayed blank.
I did a complete reset of the camera to no avail. Then I noticed the VF had a lot of dust on it, so I cleaned it - it made no difference. (I hadn't expected it to.)
But then I saw that the tiny sensor above the VF was covered in dust too. I cleaned it thoroughly and lo, everything worked as it should. Evidently the dust was enough to fool the sensor that my eye was up to it, so it wouldn't allow the monitor to come on.
I feel pretty stupid but if I can save someone else from the same mistake, I'll feel better!