If my spouse and I are deploying, but to separate bases, can we request being assigned together?

I would say it's unlikely unless you have a really persuasive reason for why. But can't hurt to ask.

The rules might be different now, but there can be a conflict of interest if either is in the others chain of command. Also if there is possibility that the two of you being on the same mission could potentially put others at risk. This would come as a loss of focus or a deeper concern over the safety of a spouse vs the entire team and mission... At the end of the day you are deployed for the needs of the services and anything that might create friction of other issues can't be tolerated. Sorry but that is the cold truth... I am sure others can explain it better and relate it to any updates to the Army rules.

As others have said, it is slim on chances but you never know unless you go to your chain of command. While I was deployed to Afghanistan, I knew a married couple with the same MOS and we were FOBbits, so they could have been stationed together but were placed on two separate bases.