At the end they are married, fun with guess your weight, coyote calls, etc. Well and good. But how did they get married? The only Justice of the Peace on the scene had no authority to perform marragies where they were, and I think he had already left anyway.
So, how did they get hitched?
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Well people thought they were in Nevada so they went to the trouble of getting a Nevada justice of the peace, once they KNEW they were in California, they were probably able to find one pretty fast and drive or fly him out there.
shareWe know they're married because it's broadcast over the radio. The announcer says they were married that morning and they 'confided' in him they were headed to South America. He goes on to say 'by now' they're probably flying over Rio de Janeiro.
From that we can deduce they returned to LA at some point; which is where they married. Returning to the ghost town shows their desire for privacy and to start their lives away from the prying eyes of the public.
Honestly, I don't think anyone cared how they got married, as long as it was clear they were married and not doing anything inappropriate.