Arr, right so it's your headunit turning on and off which is activating the screen. So nothing wrong with the screen then.
Sounds like the control circuit within the headunit is faulty. There are people who advertise on eBay that say they can repair headunits, but it's not cheap, possibly a good idea to contact one of these and see what they say. Just replacing the headunit though would be quicker but probably a lot more expensive, you're looking at around the £500 (if you can find one) and around another £100 to get it coded to your car, whereas these eBay guys charge around £350 to repair your existing one (depending on fault I'd assume).
Another option would be to see if someone would be willing just to swap over their headunit to see if that solved the problem, at least this way it would confirm what I'm suggesting.