Having a green card lets you in?
http://www.dailymail.co.uk/news/article-2800356/us-immigration-authorities-prep-order-34-million-blank-green-cards-work-authorization-papers-obama-readies-executive-order-illegal-aliens.html
The green card is a “resident Alien” card that authorizes an alien to reside here in the US. IIRC, they would use whatever their passport is coupled with the green card lets them leave and enter. I remember my wife back in the late 70s going back home with our son to visit and there was some kind of paperwork she had to fill out, but for the time she had a green card, she could travel in and out of the country. I don’t know about now.