A passport is required as the only internationally recognized form of ID. Why is anyone planning international travel without ID? Seems strange that the question would even come up. A green card means nothing to other countries, is simply an id used for residing in the country that issued it, or re-entering the country that issued it.
Get a passport then plan travel. If you were injured or became ill, and needed to seek medical attention in a foreign country, not having legal id will cause real problems.
Passports are easy and fast to get and are required.
In the case of a foreign national's passport expiring, they simply can't travel so why let it expire? I am an American living in another country, and visited 86 countries and no where, except pre 2009 agreements between Canada. US and Mexico, could anyone cross international borders without a passport. As I mentioned, it is the only id you have in other countries.
In fact, a word to the wise, level every other form of ID home, there is little or nothing in your wallet or purse, other than a passport that can be useful in a foreign country, but is a real hassle to replace if lost or stolen. If there is only downside risk, and no upside benefit why risk charging drivers licenses, SS cards, membership cards, treasured photos etc. When walking around, there is no need for the pickpocket target, the rear pocket wallet, either. Your passport, a credit card and some local currency, contact information for your ship's port agent....and that is all that you need except some personal grooming items if desired.