but isn't there still a problem with newly-generated coins contaminating many of the transactions? If some of the coins in a transaction don't "exist" on the mainline chain (or at least not in the same coinbase from the island), they can't be spent on it. (assuming the island can't make a longer chain than the Internet's chain.)
Yup, good point.
This is really an unrealistic hypothetical though. If these desert islanders have computers they have data service. If they don't have computers they'll have some local form of money and one person will do exchange and that person will have data service.
Wifi and other data services are becoming ubiquitous. Not necessarily wi-fi everywhere, but everywhere you can find wi-fi nearby somewhere.
Here's a related post - a backpacker was talking Bitcoin to chalet owners on islands in Malaysia.
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http://bitcointalk.org/index.php?topic=105464.0