DVI output only works when serial adaptor is connected

I seem to be having a really strange electrical problem with the DVI
output on both of my beagleboards. It works perfectly with the monitor
I have connected, but ONLY when the RS-232 adaptor is connected. The
moment it's unplugged, the screen turns off. When I plug it back in,
the screen turns back on...

I've had this exact same problem on both a rev. C2 and rev. C3 board.
I'd like to think it's something to do with the serial adaptor, but
the problem is that it doesn't work without it, not the other way

Has anyone else had this problem? Perhaps it's something to do with
the video cable I'm using?

Sounds to be like you have a grounding issue with your power supply. You only ground the system when the serial cable is plugged in. I would check your power supply or the socket into which it is plugged into.