Make the achievement/feat of strength account-wide and mutually exclusive.You don't get it do you?
The problem is not 25man raids doing a 10man alt-raid to get both Realm Firsts... 99% of the 25man guilds wouldn't be able to do it anyway, most alt raids of 25man guilds i know aren't able to kill more than 2-3 hcs in ToT for example.
The problem is 25man raids getting Realm First 25man in lets say week 10 and in week 11 they just take their 12-15 best players, clear the instance with overpowered setups (2 warlocks + 2 boomkins if you need it which no 10man raid could affort) and 10 ilvl more than the 10man raids to get the 10man Realm First too... there doesn't even have to be that much of a difference between the guilds... if the 25man raid is Worldrank 200 and the 10man is 300 it's already enough for the 25man raid to do so.
Only being able to get one feast per guild doesn't help either... they could just make a new guild and do it if they want to annoy the 10man raids.