When Australia and New Zealand fight, rarely does it go beyond the gravity of two siblings scrapping. Conflict between the neighbours, separated by 1,300 miles across the Tasman Sea, typically takes the form of cheap jibes based on cultural stereotypes, or competing claims to the provenance of a racehorse, pop band or meringue-based dessert. Its only violent theatre is sporting clashes, such as yesterday’s first Bledisloe Cup Test. But that, for the most part, is as far is it goes.

Over the last week, however, relations took an uncommon turn. The Australian prime minister, Malcolm Turnbull, denounced a “conspiracy” between his domestic opponents and their New Zealand allies. A tirade from Australian foreign minister Julie Bishop against the New Zealand Labour party prompted one newspaper headline to wonder, “Has Australia’s foreign minister accidentally declared war on New Zealand?”

The sniff of a diplomatic crisis arrived with the discovery that the deputy prime minister of Australia, Barnaby Joyce, is a Kiwi. Joyce, the deputy prime minister whose previous global claim to fame was a weird and protracted battle with Johnny Depp and Amber Heard over the biosecurity risk of their dogs, is a proud Australian. He was born and bred there. He leads a party called the Nationals. But his father – James Joyce, no less – was born in Dunedin in 1924, conferring on his son automatic citizenship.

For Joyce, this revelation was not just unwelcome. It was catastrophic. The Australian constitution, enacted in 1901, prohibits anyone with dual citizenship from standing for parliament. Yesterday the dual citizenship furore engulfed a seventh MP, when crossbench senator Nick Xenophon announced he has UK citizenship, thanks to his father hailing from Cyprus, then a British territory.

Joyce, who last month said of Green party senators similarly embroiled that “ignorance is not an excuse”, has renounced his New Zealand citizenship and referred his case to the high court, but resisted calls to resign. It is more than just a personal humiliation. The coalition commands a one-seat parliamentary majority. The government itself is on the precipice.

It is against this backdrop that the foreign minister launched her broadside. A New Zealand Labour parliamentarian, having had a tip-off from an Australian Labor staffer, had lodged parliamentary questions about citizenship-by-descent. Though this was apparently not what prompted authorities to confirm Joyce’s status – credit for that goes to journalists at Fairfax Australia – it was enough for Bishop to declare Wellington politicians guilty of gross intrusion. Were Labour to win next month’s election, she said, “I would find it very hard to build trust with those involved in allegations designed to undermine the government of Australia.”

In doing so, Bishop invited criticisms for escalating the matter by sticking an oar into New Zealand’s election. In the estimation of Australian Financial Review columnist Laura Tingle, Bishop had descended into a “whirlpool of hysteria and conspiracy theories that would do Donald Trump proud”.

She had the misfortune, too, to be up against a New Zealand Labour party that is suddenly being taken seriously under its new leader, Jacinda Ardern. Since her elevation to the top job just three weeks ago, Ardern has propelled the party up the polls. She chided her own MP and calmly doused down Bishop’s remonstrations. Any prospect of a full-blown diplomatic crisis swiftly fizzled out. But the case of the reluctant Kiwi and the angry Bishop had prodded at a raw nerve.

In 1973, New Zealand and Australia agreed to grant reciprocal freedom to travel and live. But that has lately come under strain, with a string of changes limiting New Zealanders’ access to social security and citizenship in Australia. Other policies, such as a hike in student fees, have led to accusations that the Anzac spirit, forged on the battlefields of the first world war, has been betrayed. Kate Hunter, a dual citizen who moved from Australia to New Zealand in 1995 and teaches Australian history at Victoria University of Wellington, says these changes are a graver threat than the Barnaby Joyce episode. “I think they have been much more corrosive of the trans-Tasman relationship ... New Zealanders feel really quite viscerally angry about that.”

In a monologue on the New Zealand current affaris television show The Project – a format licensed from its Australian masters – host Jesse Mulligan seized on the foreign minister’s remarks to highlight the plight of Kiwis in Australia. “Forget the trans-Tasman friendship in 2017 – Australia is basically a bully,” he said. “Julie Bishop – when you say you’ll find it hard to work with New Zealand, what exactly do you mean? How much worse could it possibly get?”

One of those affected by the tightening of the rules is Cal Wilson, a New Zealand comedian based in Melbourne since 2003. “Had I arrived in 2001, I would have automatically become a permanent resident, and then been able to get my citizenship,” she said. “I’m very aware that we don’t get the same rights that Aussies get when they come to New Zealand. But on the other hand, how lucky am I that I get to come to Australia, and I’m not on [a detention camp on] Manus Island? I’m in an entirely different position from other immigrants.”

The same constitution that snared Barnaby Joyce left open the door for New Zealand to join the Australian Commonwealth, a provision that the colony of New Zealand considered and rejected in the late 19th century. Part of its reluctance, says Hunter, was the implications for the Māori population. Aboriginal people had few rights, while Australia continued to use Pacific islanders as “coloured labour”.

In contrast with Australia, the crown signed a treaty with the Māori in 1840, and indigenous people gained the vote long before they did across the Tasman. But New Zealand’s record is far from unblemished, notes Hunter. The two countries’ shared legacy of colonisation includes for their native populations “shorter life expectancy and over-representation in most statistics of poverty and incarceration”.

Its sheer size means Australia has always looked down on its baby sibling. Barry Humphries satirised the relationship by giving his alter ego Dame Edna Everage a tiny, sombre and silent New Zealand bridesmaid, Madge Allsopp of Palmerston North. In their HBO sitcom, the Flight of the Conchords comedy duo struck back, painting Australians as big-headed oafs with a wide repertoire of sheep-based putdowns.

“It’s that kind of friendly rivalry, where the bigger country is really affectionate towards the small country in their derogatory teasing,” says Wilson. “There’s still the odd taxi driver who is really excited to tell me a sheep joke.” It all cuts a bit deeper for New Zealanders, however. “I think there’s an edge of fury on the Kiwi side that the Aussies don’t have. They think we’re idiots, but we’re their idiots.”

The dynamic is like the “love-hate relationship you have with your big brother”, says Bailey Mackey, producer of the reality series The GC, which followed a group of ambitious young Māori as they settled on the Gold Coast. Speaking from Sydney, where he’s following the All Blacks for a documentary, he says Australia is for many New Zealanders the land of opportunity.

“What I enjoy about the Aussies, and I’m going to steal something my Australian mate who’s sitting beside me here told me, is that their coat of arms says it all. They have the kangaroo and the emu, the only two animals in the country that can’t move backwards. That sums up the Australian mentality. They really are a front-foot country.”

Are New Zealanders, then, nocturnal, flightless birds? “We’re probably a bit more subtle. I think we tend to watch a bit more, and take things in, before taking action or talking.”

Back in the Australian parliament, poor Joyce was greeted with sheep jokes. “Baaaar-naby,” jeered opposition MPs. One asked: “If the foreign minister won’t be able to work with the New Zealanders, how will the foreign minister be able to work with the deputy prime minister?” Another hollered: “Vote Jacinda!” Even Amber Heard exacted revenge for the deported dogs, announcing she’d sent him – “assuming this passes his biosecurity laws” – a case of kiwi fruit.