Work for Japan company with SOFA status?

I am currently in Japan under SOFA status as a dependent for my husband who is in the USMC. I have been offered a job with the company I worked for in the US, but this is under their Japan entity. Does my SOFA status cover me to work for a private Japanese company?

I was told that I would have to denounce my SOFA status if they were to sponsor me with a work visa.

If the answer is yes, you can work for a Japanese company with your SOFA status, what is they way you show identification/proof if there is a need for it?

Hmmm... I vaguely remembered that this question had been asked before, and it has:
https://www.japan-guide.com/forum/quereadisplay.html?0+172827
But the reply was ranging from yes to no. However asking the legal advise on base kind of sounds like a good starting point.

I would be asking either Immigration or the base for the specific details, just to make sure you get the right answer.