My fiancé is in the Navy and we plan on getting married in June but afterward we, my son (his step-son) and I plan on moving to Japan where he is station.

What kind of passport will we have to get? Can he sign for my son's passport after we get married?

I've heard of a non-fee passport but I'm not sure of what a non-passport is and if we are even eligible for them.

The no fee passport is for diplomatic travel, which I don't think you qualify for. You and your son will have to have regular passports from whatever country you live in.

You are responsible for your son's passport, no one has to sign for it.

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If you are US citizen, you and your son just carry US passport like everyone else. If you are non-resident or permanent resident (greencad), you carry passport issued from your home country.