You book the ticket in the name of your passport. You show your passport when you leave the US, and you show your passport plus the Green Card plus the marriage certificate (only if they give you a hard time) when you leave Peru. In the US you show your Green Card and your passport.
Your passport is the document that allows you to travel internationally. The Green Card is the document that allows you back in to the United States.
RiverJourney reacted to Brother Hesekiel in Traveling Green Card and passport different name