@Matthew Crawford just like @Michael J. said - most things can be done electronically, and if you have a good team here, they can handle the inspections, etc for you!
But if the lender/title company needs 'inked' signatures - You may need to go to an Embassy for signing (if lender required but seems a little extreme), but a notary should be able to witness it as well (they do all the time for out of state contracts here)- like at Pentagon Federal (or the Community Bank there on base - I'm not sure what base you are at, but that was the one on Kadena I think!). Then you'll just 'overnight' it back to the states - (several days to a week in reality).
So, the title company will send you electronic copies you will print out, (you will have to pre-sign a few days ahead of closing more than likely), get witnesses by a notary, 'overnight it', and when your inked signatures are back stateside, they will fund the loan.
We were stationed in Okinawa for a couple of years - LOVED it - but it is a pain to try to do business with the time difference. I have an investor client in Okinawa right now, and it can be several days delay between questions/answers/back and forth - which isn't idea, but still doable. Best of luck - let me know if you have other questions I might be able to help with!