Chris -
Without the second PC attached to the monitor (LCD), power cycling the monitor (or allowing it to enter a sleep state) does indeed cause an offset to appear/disappear. This offset appears to be dependent upon the monitor (which one it is, versus where it's plugged in power-wise). The only solution seems to be to make sure the offset never changes, by either keeping the monitor always on or always off. The latter wasn't an option on this machine.
I agree that it's quite bizarre, but somehow it's working fine now by keeping the monitor constantly on ...
Brett