The SSD Executor does not have an Officers' Mess – how could she, with more than ten thousand officers aboard.
There are Officers' Messes No. 1-24, though. While theoretically open to anyone above a certain rank, three of those are firmly in the hand of the Army contingent aboard, and it would be inadvisable for any Fleet officer to intrude. Likewise, Army officers can expect a cool welcome in any of the remaining ones.
At a certain level of rank, however, officers become increasingly thin on the ground. The Officers' Lounge, domain of the highest field grades and above, has its invisible demarcation lines crisscrossing the room itself.
Though not all of those fall along the Army/Navy divide...