What kind of workout attire is allowed at Al Asad Airbase Gym?

it depends on your unit SOP or the base SOP once there. Units can always have their own specific SOPs. For example, when I was deployed to Iraq first time, our BN CSM didn't allow us to wear PTs anywhere except during PT or if you were going to the shower. We couldn't wear them in the DFAC but other units could.

There is generally a unit or installation policy that will dictate. Stuff like that changes regularly depending on the personalities of those in charge at any given time. While at Bagram in Afghanistan it changed almost every week or so for a while because of brass getting spun up on what they saw people wearing.

It is up to the Command- ask them