As others have said, it depends on what other options are available. I can tell you it's not followed to a "t". While in ROTC, I was accepted to do a "Cultural Immersion Program" to Turkey, run through Air Force ROTC. The group met up at JFK and then we all flew Turkish Airlines from JFK to Istanbul, despite the fact that Delta also ran a nonstop JFK-IST flight. A cadet from my detachment went to Turkey on the same program the following year. They flew AF out of BOS and connected in Paris before continuing on to Istanbul. It was the same deal for cadets I knew who went to other countries - they usually flew an airline of the host nation (the group that went to India flew Air India, the group that went to South Africa flew SAA, etc.). Perhaps that was part of the deal between the Air Force and the host nations for the program I went on was that we flew their carrier.

On active duty I've seen the similar events. My first assignment on active duty was working on the F-35 Joint Strike Fighter and a couple people from my office had to go to Turkey. Again, despite the fact they could get to Istanbul on a US carrier via JFK on DL, they flew UA to Munich and switched to LH to get to Turkey. Not sure if their flight was coded as a United flight or not though.