Jacinda Ardern's election makes things awkward for Julie Bishop

The elevation of Jacinda Ardern to the New Zealand prime ministership electrified pundits on social media, who predicted awkward times ahead for Foreign Minister Julie Bishop.

In the midst of the Barnaby Joyce citizenship crisis in August, Ms Bishop declared she would "find it very difficult to build trust" with New Zealand's Labour party following revelations that a Australian Labor figure made contact with a NZ Labour MP.

Ms Ardern returned fire, calling Ms Bishop's "false claims" were "highly regrettable".

"I greatly value New Zealand's relationship with the Australian government. I will not let false claims stand in the way of that relationship," she said.

"I would happily take a call from Julie Bishop to clarify matters. I have also contacted the Australian High Commission to register my disappointment and will be meeting with the High Commissioner later today."

Their skirmish prompted a barrage of comments on social media on Thursday following NZ First leader Winston Peters' shock announcement that he was forming a coalition government with Ms Ardern's Labour party.

Some commentators dished up sarcasm; others criticised Ms Bishop including Sky News' conservative pundits Paul Murray.

Mr Murray called Ms Bishop's August remarks a "disgraceful overreaction" and "even more ridiculous tonight".

Australia's Labor politicians also weighed in to suggest Ms Bishop had bridges to mend.

Labor's Penny Wong said it was incumbent on Ms Bishop to "undo the damage caused by her irresponsible remarks". It was Senator Wong's chief of staff who sparked the NZ Labour questioning that led to the revelations about Mr Joyce's citizenship status.

"How would it feel to be Julie Bishop right now?" asked fellow Labor frontbencher Tony Burke.

Asked by a journalist if she could trust Ms Ardern, the Foreign Minister said: "The fact is Ms Ardern gave an explanation a couple of months ago about the behaviour of a New Zealand member of the Labour party and she said at the time that his conduct is wrong, it was unacceptable and it should never have happened and that he shouldn't have become involved.

"And I accepted her explanation and I agreed with her absolutely."

Ms Bishop also delivered a withering response on social media, attacking the national affairs editor of The Courier-Mail, who said she "might have some fences to mend".

She also labelled as "rubbish" commentary from the ABC's Jonathan Green and Fairfax Media's Peter Martin.