Thanks for that, I've solved the problem from reading the manual! (shock!)
I tried the Communication On setting before, but now I know that the AV Output needs to be Always On too. For anyone in the future that that has this issue, here is how to resolve it.
Menu > Extended > AV Settings > AV Output > Always On
Menu > ECO > Standby Mode > Communication On
Both tasks require the projector to be turned off for the setting to take effect, then your Monitor Out should stay on.