Thanks for all the awesomeness Mojang!
I created a Realms game, and it was set to Normal when created.
(My own game mode was on Easy which I didn't notice until I discovered the problems).
When I joined the world I found it was on peaceful mode. Hostile mobs appear and despawn immediately. You can only glimpse them for a brief flash. No damage except falling which heals immediately, and no hunger. I double checked the config menu but it still says the Realms world is set to Normal.
I invited someone to join a while later who had their personal game mode on peaceful, so I got them to change it to Normal, and changed my own game mode to Normal in options while connected to the realms map, still no mobs and no hunger. Logging in and out makes no change, nor did changing the realms difficulty level to another level and back to normal again. I can't change the Realms game mode to Creative either.