An electronic mystery -- any ideas?

When I turn on the desktop computer in my office, the monitor doesn't come on. I've discovered that what I have to do is, after turning on the computer, unplug the monitor and plug it back in in a different outlet.

Suppose the monitor is plugged into Outlet A when I turn on the computer; I have to unplug it and put it in Outlet B before it will work. I use the computer for a while and then turn it off. Next time I turn it on, the monitor is already plugged into Outlet B, but now it won't work until I unplug it and put it in Outlet A. So both outlets work equally well; the key point is that I need to change the outlet after turning on the computer. Changing it before turning on the computer doesn't work. Neither does unplugging the monitor and plugging it back into the same outlet again. Very occasionally, I have to move it to a third outlet before it will work -- but again, which specific outlet is the third doesn't seem to make any difference.

The monitor itself is not the problem. Other monitors used with this computer exhibit the same strange behavior. This monitor used with other computers behaves normally. All the outlets work completely normally if I plug anything other than a monitor into them.

I've had a computer technician and an electrician look at the problem. Neither of them has a clue what's going on.I'm posting this here on the off chance that any of my readers can solve the mystery.