I'm thinking Imperium takes it, basing it solely off of sheer numbers of ground military forces as well as spaceborn assets. The sheer number of troops available to the Imperium, combined with the massive number of transport ships, added in to the small number of worlds in the Earth Alliance, means they can literally swamp every single world utterly.
I have no idea about the power of their individual ships, though. Imperial ships, for instance, have a normal output in the terraton to petaton range, with their weaker weapons in the dozens to hundreds of gigaton range. How much damage do the weapons on Earth Alliance fleets inflict? If they only inflict damage in the gigatons, max, then they're hosed (likewise, one would assume that they can take about as much damage as they can put out).
Seems to me that the Earth Alliance is a power which can be beaten by the Federation. And if the Federation can beat them, then the Imperium can certainly beat them. They don't even have shields, so whereas teleporting Terminators aboard a Starfleet ship wouldn't work, it would work against an Earth Alliance ship. And Earth Alliance hand weapons aren't anywhere near as powerful as phasers, so it's doubtful if they'd even be able to penetrate Terminator armor or their built-in force field, making a five-man squad pretty much unstoppable.
Likewise, Imperial ships have both bombers and fighters, and if their bombers are powerful enough to inflict heavy damage on a 40k ship, then they could be nightmarish to an EA ship. Altogether, this is an even more vicious curbstomp then what the Federation would experience against the Imperium.