Forget what I said about Emu being shy: it was all an act.
He's now become the more boisterous of the two, and is normally the instigator of any scuffles, despite being significantly smaller. He's not really been under the bed lately, and is currently curled up on my lap.
After all my litter box dilemmas, I've decided we need to try one with a lid - of the two boxes we have, they both have a strong preference for the one in the en-suite bathroom. Unfortunately, they also have smelly poo, and this is not ideal when they need to go in the middle of the night.
Emu also doesn't bury his poo, which I suspect is him trying to assert his territory.
So, I've just bought a hooded litter box and put it in the en-suite instead (same litter, as I don't want to freak them out too much - and I've left the flap up). We shall see what happens.