Rainbow Warrior type incident happens between the US and Australia.
That'd be all it takes. Australia isn't a part of NATO so just having chilly relations with the US would make it non-aligned for all intents and purposes, because the US is the only western power there with any clout after the 1960s or so when Britain withdraws from the region. Bonus points if a Republic referendum succeeds and they adopt the Golden Wattle flag, and there you have a nondescript, non aligned, antipodean Republic that occasionally shows up in Eurovision.