Would it be possible to get a South American theater in WWI? Could, for example, Argentina join the Central Powers in order to take the Falkland Islands and then for some reason have Chile join the Entente? Any other ideas are welcome.
If there was a Brazil, Venezuela, and possibly Chile vs Argentina and possibly Bolivia (to retake their coastline) front, how would that change the war?You got that reversed. Argentina was practically a British colony in this period and very nearly joined WW1 on the Allies side, for reasons similar to Brazil. Meanwhile, Chile's military was under heavy German influence (although, like Japan, its navy leaned British, and it had also a large number of citizens of British descent).
Getting a South American front that is not just a local conflict subsumed into the greater war is going to need an old POD. Venezuela and Brazil are the only countries with believable disputes against Entente powers, but Venezuela wouldn't go at it alone. As for Brazil, I think you need to keep the Empire going and the country pursue an international, non-American expansion by the end of the century, like the United States, instead of the OTL retreat into isolationism and the conflicts with its own navy that followed the overthrowing of Pedro II (the navy was a hotbed of monarchists).
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If there was a Brazil, Venezuela, and possibly Chile vs Argentina and possibly Bolivia (to retake their coastline) front, how would that change the war?