Anthony Albanese loves a trophy, especially a human one. He prides himself on his various “captain’s pick” candidates – good campaigners he has steered into seats.

Way back in the Gillard days, he was key in persuading discontented Liberal Peter Slipper to defect. Slipper became an independent and Labor’s speaker.

The exercise helped the government’s numbers, but the bold play didn’t end well for Labor or for Slipper. The government was tarnished, and Slipper, relentlessly pursued by the Coalition and mired in controversy, eventually had to quit the speakership. The affair did produce Julia Gillard’s famous misogyny speech, however.

Now Albanese has another gee-whiz prize – Western Australian Senator Dorinda Cox, who has defected from the Greens. Cox, after being defeated in a bid for Greens deputy leader, approached Labor and the PM drove her course to being accepted into the party.

The manoeuvre makes a marginal but insignificant difference to Senate numbers – Labor will still need the Greens to pass legislation opposed by the Coalition.

Taking in Cox is a risk, and some in Labor are looking at it askance.

The prime minister’s embrace of Cox contradicts Labor’s argument when its Western Australian senator Fatima Payman defected to become an independent. It said then hers was a Labor seat and she should therefore resign. But this wouldn’t be the first time expediency trumped consistency in politics.

Cox, who is Indigenous and was spokeswoman for First Nations and resources in the last parliament, has been a fierce critic of extending the North West Shelf gas project, which the government has just announced. Albanese says he is confident she “understands that being a member of the Labor Party means that she will support positions that are made by the Labor Party”.

She has also faced allegations of treating staff badly. Labor discounts the claims against her, saying they are overblown and a product of Greens factionalism and toxicity. Certainly, she was given a tough time by the hard-left faction represented by deputy leader Mehreen Faruqi. Labor would be wise to ensure Cox feels supported in her new party home.

Albanese perhaps calculates that the worst that can happen is there’s a blow up and she defects to the crossbench. Labor could shrug and say, she was never really one of us.

Snatching a senator from the Greens is particularly satisfying to Albanese because he hates the party so much. Last term, lower house Greens MP Max Chandler-Mather (defeated at the election) really got under his skin. More generally, the Greens held up important legislation, most notably on housing.

In the new Senate, Labor will need only the Greens to pass legislation opposed by the Coalition. How new Greens leader Larissa Waters – who replaced Adam Bandt after he lost his seat – handles the party’s relationship with the government will be crucial for the more contentious parts of Labor’s legislative program.

The usually low-key Waters will be under a lot of pressure. The Greens had a bad election, losing three lower house seats. Now they have lost a senator at the start of Waters’ watch.

Waters conceded on the Serious Danger podcast in late May that Labor had successfully run the narrative of the Greens as blockers. “So I do think we’re going to need to be quite deft in how we handle balance of power in this term, […] People want us to be constructive. They don’t just want us to roll over and tick off on any old shit. They want meaningful reforms.”

Waters will want to pick her fights carefully, and also find ways of pursuing the Greens’ agenda where the party co-operates. The first deal is likely to be on the government’s legislation to increase the tax on those with large superannuation balances, which contains the controversial provision to tax unrealised capital gains.

Opposition Leader Sussan Ley and her team will confront some of the same problems as the Greens – when to oppose and when to seek to negotiate with the government.

For his part, Albanese will have a novel challenge with Ley – what stance to adopt against the first female opposition leader, especially but not only in parliamentary clashes.

After facing two alpha male Liberal leaders, Scott Morrison and Peter Dutton, a new approach will obviously be necessary. As one Labor man succinctly puts it, “Labor can’t monster a woman”. There can be no repeat of Albanese, a frontbencher a decade ago in the Shorten opposition, interjecting to urge a female colleague engaged in a stoush with Ley to “smash her”.

For Ley, trying to deal with the Liberals’ multiple difficulties in attracting women voters and candidates must be high on her agenda. Former Liberal federal president Alan Stockdale, one of the three-person group currently running the NSW division of the party, showed himself part of the problem when this week he told the NSW Liberal Women’s Council, “The women in this party are so assertive now that we may need some special rules for men to get them pre-selected”.

Stockdale said later he was being “light-hearted”. Tone deaf might be a better term. Ley jumped on him. “There is nothing wrong with being an assertive woman. In fact I encourage assertive women to join the Liberal Party.”

The jury is out on whether Ley will be able to make any sort of fist of her near-impossible job. But in the short time she’s been leader, she has shown she is willing to be assertive.

She emerged from the brief split in the Coalition looking much steadier than Nationals leader David Littleproud, even though she had to persuade her party room to accept the minor party’s policy demands.

In her frontbench reshuffle, she was willing to wear the inevitable criticism that came with dropping a couple of senior women who had under-performed.

As deputy leader, Ley adjusted her style a while before the election, toning down the aggression and sometimes wild attacks, that had characterised her performance earlier in the term. A Liberal source said she found her “line and length”. As leader, she will have others, notably deputy Ted O'Brien, to do the head-kicking, giving her room to attempt to develop a positive political persona.

Labor leaned into attacking Dutton – never afraid to name him. With Ley, Albanese might adopt the Bob Carr approach of avoiding using his opponent’s name. At least until he finds his line and length in dealing with her.