They used Iroquois (Haudenosaunee) visas. To put it bluntly, countries like the U.S., Canada and South Africa that have "sovereign lands" arrangements with pre-European inhabitants are playing a game. We relax a few laws and call you a "sovereign" nation. It works fine, as long as there is no one involved besides the "sovereign" government and the surrounding, real government, and everyone plays the same game of pretend. It seems especially ridiculous when you consider that there are plenty of Native Americans who have no Federally recognized status as such just because they've never been "Federally recognized". That is, your Indian nation only exists if the U.S. government says it does. (In Northern California, Wintus are still not recognized, whereas the Feds did recognize some business partners who wanted to open a casino in San Pablo, CA.)
Where the rubber meets the road is when something happens that requires a foreign government not playing this game to recognize their sovereignty - as happened here. This would be bad enough even if people hadn't previously entered the U.K. using a Star Trek Federation visa (if you have the link, send it.)
That being the case, if American Indian reservations want to be recognized internationally, they have to start behaving like actual sovereign nations, and this means (for one thing) if you want to use your own travel documents, establish the relationship ahead of time, instead of assuming that the U.S. will do it for you. I've often wondered why Indian reservations in the U.S. don't push the issue more, and experiment with more than just gambling laws. Anecdotally, it certainly seems to me that gambling on rural reservations isn't contributing positively to the trade balance with the outside world; it seems that the vast majority of people gambling are from the rez, and the only wealth movement (not creation) is going from the pockets of people on the rez to the few people on the rez that own the casinos. I recall a case in the 80s or 90s of a reservation in Oklahoma that was pissing off the Federal government by offering wacky interest rates on loans and bank accounts, though I don't remember the details and can't find it now.
The real reason to support reservations pushing the issue with the Federal government is not to create anarchy but because it would provide a great opportunity to improve our democracy; there would be pressure on Federal institutions, and the closest thing to competition that government has. Why not more and more dramatic experiments than just gambling? Drug decriminalization is an obvious one, but what about foreign relations (begin relations with China and see how the Feds like that), medical research, numbered bank accounts, various internet safe-havens? It would be pretty awesome if the Navajo Nation offered its own Phase I drug research facilities. If that sounds potentially exploitive, you can mandate a certain level of investment in the local economy or a certain number of local employees like Thailand did when every corporation and his brother started doing HIV research there. And I'm sure lots of people (especially foreign pharmaceutical personnel with lots of currency!) certainly wouldn't mind a work-related excuse to be in the proximity of Canyon de Chelly and Monument Valley.