Okay, I'm obviously not a SOFA expert.
But then why would one have to pay attention to things the way you describe in your previous post (#7)? If it's in support of the US mission, so SOFA, why would that record-keeping matter (or doing things on/off base, etc)?
If a soldier, or contractor, or their dependents do base/mission work, that should be SOFA, regardless of location.
But any of those same people worked on the local economy (e.g., some Eikawa lessons), I'd expect that to be taxed by japan. Similarly, if one of those same people were teaching Eikawa remotely, I'd expect this also to be subject to japan taxes--it's unrelated to the SOFA mission (whether it's done on or off base).
Again, not likely something that the NTA would be too interested in.
If SOFA status does kind of shield it from japan taxation, is it then an IRS thing? Social security deductions, income taxes, or anything else on the US side that might apply (being a contractor and 1099'd, or an employee and getting a W-2)?