None of the current frax(that are based in the US) will let you be based outside of the US. I doubt any of them would even hire you if you told them you wanted to live in Belize, but would get yourself to XXX gateway city in the US. It's too much of a hassle, too much chance for something to go awry with you getting to work, and there's plenty of pilots out there looking for the same job.
At Options, we can airline out of domicile if we provide 48 hours notice. This means that if a pilot is based at LAX, he can, at the end of his tour, have the company airline him to JFK instead of LAX. They will then airline you back out of that location as well, if necessary. No cost to the pilot...but they won't allow anything outside the US as a general rule...and I haven't tried it, but I doubt they would allow Alaska or Hawaii due to the high cost.