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Ok. They can't process the BAH pay until he is out of the barracks, but the BAH should be backdated to the date that you were married, because it is for your support as well as his. It isn't just for him to move out of the barracks.
It isn't going to be quick about coming either. The first month you get it will likely just have that month's BAH. It could be two or three months before they back date it and pay it out. For that time that he lived in the barracks, they will sometimes just pay a "differential" of what a single person's BAH would be and one with dependents, but I can't tell you if they WILL do that, just that they can do it.
The difference between the two. The single person's BAH is what is paid to him for his housing. The dependent rate is the additional funds for your upkeep.
If they seek to pay the differential, then his living in the barracks (technically, by having a room) means that they don't have to pay his upkeep, just yours.
So, around $300 for that backdated BAH during the months that you were married but he still occupied barracks space.
But, they don't always do that. It really is up to the commanders and NCO's that submit the paperwork and how the personnel office processes it. Legally, he's only entitled to the differential, but many times he'll get the full BAH backdated to your marriage date. I just can't promise that, because I don't work in that personnel office.