I've currently got my X-33 up for sale. I've been in two minds whether or not to let it go anyway, and then to throw a curveball, somebody interested in the watch has told me that the "issued" X-33s generally come with a card from Omega, Bienne, which mine does. There are no markings on the case-back exterior. I haven't removed the case-back to see if there is anything on the underside though. I'm wondering if it's worth paying for archived paperwork from Omega or not. This is the watch and card...
I'm not sure that you can get an extract on a watch that new, you're right in that generally the special request engraved caseback models are stamped Bienne, but they can also be marked as such just because someone important requested one from the factory.
From what I have read, Omega at one time offered a discount program on the x33 to active duty military personnel, with a minimum order size. Those watches so purchased would typically have a warranty card issued in Bienne. These watches are not military-issued, but instead were bought with personal funds. Some have engravings on the back as they were ordered for a particular unit. They don't seem to command any premium in price, unlike many watches purchased by a government for use in the military. Here's an example, credit to flashharry on mwr.
Think it is 20-25 plus watches to get clear backs and engraving is done by third party. Can be divers also. Read it somewhere a while ago.
These X33 are not "issue" watches. It's a marketing / sales offer from Omega to military pilots / units. They were able to buy the x33 for approx USD 600 for the Gen 1 (more later) and get the case back engraved. You can find a lot of these watches sold on eBay. Not too much of a premium to get one. Some engraving are nice and unusual, some - like the F15 - are quite common.