duffdouglas said:
We're currently in China and the issue is the Argentine embassy here doesn't recognize her passport and have said she has to go to Taiwan to get it processed. Ok, fair enough. BUT, when she gets her tourist visa and wants to extend it after three months I forsee issues. Going to Uruguay probably won't be an option and in reality flying back to Taiwan would be a pain and fiscally just expensive.
If I have a work visa does it make a difference for her (if hypothetically we were married?)
I do not have experience with Taiwanese citizens requesting visas for Argentina. There is, however, a significant Taiwanese community here. The Ministry of Foreign Affairs page says that they require a "certificado de viaje" (travel certificate). I suppose it is similar to the authorization that has to be required for Chinese citizens, though I'm not sure. The issue here is that Argentina does not recognize Taiwan's independence.
To get it processed, she will have to go to the Argentine consulate with jurisdiction over Taiwan. I looked it up online but couldn't find it; maybe in the
Argentine consulate in Hong Kong they will be able to give you more information.
Once she obtains the tourist visa, she will be able to renew it once, for a maximum of 90 days, at the Immigration agency, without having to leave the country. If you're married, she may apply for a visa which will be granted for the same amount of time as yours.