Okay so the specifics of the actual game played here are far less unusual than the other examples given, but I think it belongs: international rules football. I'm going to have no choice but to refer to what I would normally call "football" without a qualifier as soccer here, so don't let any instances of British English trip you up.

Ireland and Australian both have local ball games, Gaelic football and Australian rules football, that are popular in that country but not big anywhere else. Neither is developed from the other, but they have a lot of distinctive commonalities: you can carry the ball, but only a limited distance before you must bounce, kick, or pass it; you're allowed more contact than soccer, but less than rugby or American football; there are two ways to score, an easy low-point option and a difficult high-point one; and they are played on very large pitches compared to those sports. Both games are very fast-moving and free-flowing, eschewing the structures of rugby or American football for something more like soccer. Both are good fun to watch, I recommend them if you enjoy other kinds of football.

Back in the 60s, an Australian and an Irishman living in Australia had the idea to get an Aussie rules team together and do a tour to Ireland, plus the UK and US where there were big Irish communities. They played Gaelic rules with modification to accomodate the Australian players.

Fast forward to the 80s and someone reckons this should be a proper thing, so international rules are developed - a proper hybrid of the two that attempts to be as fair as it can to players of both sports. The goals are a combination of both sports' goals, the ball is round and the pitch is rectangular like in Gaelic but the tackling and kick-catching rules are Australian. And then once every couple of years, Ireland and Australia play a game against each other.

Of course, both sides play their own game the rest of the time. So it's a sport that is almost nobody plays except for international tours between Ireland and Australia.