Yes, they do! At least over here in South Korea.
Before arriving to Korea last year, I began researching the licensing procedures. I have known fellow soldiers who have served over here in Korea and were able to get an HL callsign and get on the air. By reviewing an old Army regulation that was specific to Korea, I saw that there was a process where a soldier could submit a memorandum requesting to be issued an HL9 prefixed callsign. The soldier is required to already possess an FCC amateur license. However, a problem emerged. No one knew who to submit the request to. There are a number of American hams over here, most who work in some aspect for the US military. These individuals generally do not qualify for an HL9 callsign and therefore the general knowledge of how to go about applying for these HL9 licenses was lost. The process to apply for a standard South Korean amateur radio license is straight forward and is what I did. I was subsequently issued the callsign HL2/AD7MI, with the “2” in HL2 signifying the geographical area where I am stationed. The callsign got me on the air, but I soon learned the difficulties in using such an unwieldy callsign (“Hotel Lima Two Stroke Alpha Delta Seven Mike India” is a mouthful).
The good news – an Air Force ham, who is stationed down at the US military headquarters in Seoul, kept plugging away at the HL9 issue and was successful in not only getting issued an HL9 callsign but also was able to define the process for the rest of us. Finally having an organization identified to submit my request to, I was issued my new callsign: HL9MI. I am looking forward to getting on the air and breaking it in.
As far as I know, Korea is the only place where the US military issues callsigns. Back after WWII, it was a common practice but it was discontinued in Japan and Germany once their amateur radio programs stood back up with their civilian governments. Amateur radio licenses for US military in Germany is issued through the US military (at least that is how it worked in 2002), but that program is run by the German government. Even in Iraq, licenses were initially issued by the Coalition Provisional Authority before the responsibility was transferred to the Iraqi civilian government. The Afghan government is responsible for the issuing of callsigns for NATO military serving in Afghanistan. It is interesting that here in Korea, 60 years after the civilian government of South Korea stood up, there is still an active provision for the US military to issue amateur radio callsigns.