Perhaps a far left wing New Zealand, and a very unpopular right wing government of Australia.
Have the Anzac armies left leaning, and launch a coup against the Australian Government during the interwar years. Even better, during WW1?
A couple of other ideas- Marxist unionists overthrow New Zealand's government between 1912-1913. As a result relations between the UK and NZ become very tense, massive reforms in NZ are blocked almost every time by the imperial parliament, but they are carried out and passed into law anyway. To allow these reforms to continue, the judiciary is essentially purged of pro-Imperial judges.
Australia becomes very tense as it's own unionists start to follow their peers in New Zealand. A very unpopular Fisher government collapses as civil disorder spreads (unlikely I know). A more right wing government comes to power. In effort to quell the civil disorder they decide to ship out as many "trouble-makers" (unionists) to war, or put them in prisons, then into camps as the prisons fill up.
New Zealand refuses to send troops to war, but eventually capitulates, and decides to send a token force. A home guard is set up to replace the armed forces overseas as the Army at that stage is not trusted. Around this time the first ANZAC is formed. Leftist Australian troops help convince the kiwis that their cause is best.
After the war, and the horrific losses, the governments of both Australia and New Zealand are very weak. They are blown away in elections. New Zealand becomes more centrist, Australia more leftist. The Australian government is eventually dominated by ex-Soldiers and union leaders. It evolves into an Anti-British military government as civil liberties and imperial powers are discarded.
When the Military government actually becomes formal New Zealand, now much more moderate faces a return to the unrest when the unionists first took power. The different factions of the armed forces and unionists clash. Australia lends a great deal of (mostly political) support to the leftist sides, and eventually the NZ government is toppled and replaced with a pro-Australian ANZAC led government that surrenders power to Sydney.
Mostly fantasy I know. Any thoughts?
(Im a little sketchy on the Aussie politics, when were the unions most militant?)