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The green card is evidence of US permanent residency. It is an immigration document, not a travel document. You would still need a valid passport to travel. Now as you're still an Omani citizen, you could likely get an updated Omani passport through your embassy or nearest consulate (if you don't have an up to date passport). But you can't simply travel out of country and back with your green card. You'd need a passport to do so. Even with a green card, you couldn't get back into the U.S. without a passport. Again, it's an immigration document, not a travel document, and you'd need the passport to be able to travel.
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