When Theresa May boarded an RAF Voyager plane at Heathrow last Thursday afternoon and headed for a two-day G7 summit in Quebec, she must have been hoping for at least a little respite from arguments with her cabinet over Brexit. She had, in any case, to use some time in the air to prepare for what promised to be a difficult meeting with fellow world leaders, in a number of respects.

President Trump had reportedly been referring disparagingly to May’s “schoolmarm” style and political correctness in the run-up to the meeting. More serious arguments were also looming with the president, the most pressing being over his decision to impose steel tariffs on the US’s European and other allies, risking a trade war.

But Brexit still stalked her all the way across the Atlantic. First there was the wash-up from a heated row with her Brexit secretary, David Davis, whom she had to talk out of resigning over a “backstop” deal on customs before she headed to Heathrow. That dominated a press briefing mid-flight. Then, as the plane was taxiing into the airport terminal at Bagotville airbase, near Quebec, news reached the UK contingent that the foreign secretary, Boris Johnson, had exploded another bomb under cabinet collective responsibility. Johnson had been secretly taped telling a private meeting that, under May’s leadership, Brexit negotiations were heading for “meltdown”. Trump, he said, would do a much better job.

Journalists demanded an immediate response from the PM, who had heard nothing of Johnson’s outburst while in the air. Would she sack him? Did she still have full confidence in her foreign secretary? The questions were thrown at the prime minister’s aides as she switched planes onto a Canadian Air Force Hercules and headed to the summit at the Manoir Richelieu resort, leaving the press on the tarmac.

As MPs prepare to vote in the Commons this week on crucial Brexit amendments passed in the House of Lords, and as an EU summit approaches that is supposed to agree a post-Brexit customs plans to solve the Irish border problem, any pretence of government unity has been abandoned. The two sides of the Tory Brexit war argue openly, whether they are rivals in cabinet or on the backbenches. It was not only Davis who was threatening to resign on Thursday but several other Brexit-supporting ministers too, among them Davis’s deputy Steve Baker. They – like Johnson – were furious that May’s latest customs plan did not put a time limit on the “backstop” proposal for keeping the UK aligned to the EU customs union if it fails to reach a deal.

That omission looked to Brexiters like a ploy to tie the UK permanently into parts of the EU economic system, and a victory for Tory Remainers. May had invited Remain supporters including Anna Soubry and Ken Clarke into Downing Street earlier in the week to brief them on the backstop plan (with no end date). Unsurprisingly, they loved it. Davis’s reaction, equally inevitably, was precisely the reverse. He demanded to see May on Thursday morning and told her in no uncertain terms in two private meetings that she had to include an end date, or it would look like Brexit was never going to happen.

“It was the most frank discussion they have ever had,” said a Davis ally. “They have not talked like that before. He made it clear that he had to be listened to if he was to carry on as Brexit secretary.” The PM and Davis then agreed a fudge that said the government “expects” the backstop to conclude at the end of 2021. As one middle-ranking minister put it that day: “All the PM’s energy goes on trying to keep the peace. She tells one side what it wants to hear, then reassures the other side. So we go nowhere.”

Veteran pro-European Ken Clarke was briefed at No 10 on the customs ‘backstop’ plan. Photograph: David Levene/The Observer

The disputes run wider, into the heart of the Westminster machine, preventing the joined-up government that is essential in negotiations as complex as those about Brexit. Business leaders who meet ministers in Downing Street say they are listened to politely but never get clarity. Davis is said to see the hand of Olly Robbins, May’s chief Brexit adviser at No 10, behind a drift towards soft-Brexit thinking in Downing Street. Robbins was moved from Davis’s department last year, where he was permanent secretary, as the two never got on. The PM now leans on Robbins increasingly. The trouble for her is that there is increasing speculation in Whitehall that Robbins has had enough and is plotting a move to the private sector by the autumn. While this is denied by No 10, such gossip in high places does not inspire confidence that all is well.

The big question now, with the executive in turmoil, is whether parliament will begin to take control – whether a legislature with a large Remain majority will start to seize the initiative in the Brexit process this week.

On Tuesday and Wednesday, MPs will debate and vote on a string of amendments passed in the Lords, several of which would wreak havoc with May’s Brexit strategy and further damage her authority if passed in the Commons. They range from one which calls for the government to take steps to negotiate to stay in the customs union, to another which urges it to work towards membership of the European Economic Area (EEA), which would allow the UK to remain in the single market after Brexit.

True Remainers want both outcomes, but on the Tory side there is much agonising about how best to get there and whether this week is the moment to strike. Both Tory and Labour MPs who oppose hard Brexit are furious that Jeremy Corbyn has decided to order his MPs to abstain rather than back the EEA amendment, when there were, possibly, enough Tory rebels to support it had Labour piled in.

Hilary Benn, who is one of dozens of Labour MPs who will defy his own leader and vote for the EEA amendment, as well as for the customs unions plan (which Labour does support), says this is the moment.

“It’s a chance for parliament to take control amid the evident chaos inside the cabinet, which is worrying business and undermining future investment in jobs and prosperity. If government can’t lead, then parliament must,” Benn says.

On the customs union, pro-EU Tory rebels seem divided this weekend. Clarke and Soubry are likely to stand firm and rebel – voting against the government in favour of the customs union plan – and it is not impossible that enough others may join them for it to be passed. But at least 12 Tories would need to defy their own government, and the “rebel dozen” are split on tactics. If any fall by the wayside, the chance of defeating the government is gone.

‘It’s a chance for parliament to take control,’ says Labour’s Hilary Benn. Photograph: Christopher Thomond/The Observer

Some wonder whether defeating May now, when she is moving in their direction on wider Brexit thinking, might be self-defeating, as it would unleash leadership threats from the Brexiter hardliners just when Remainers had got May where they wanted her.

“It is a political calculation,” said one leading Tory Remainer. “If we were to defeat her on that now, does that further weaken her and give the ERG [the hard-Brexit European Research Group, led by Jacob Rees-Mogg] more opportunities to stick their knife into her? And perhaps that is not where we want to be.” This group may decide that it would be better to vote for a stronger amendment on the customs union and on the EEA when the customs and trade bills return to the Commons in July – when it would be too close to the summer break (and Brexit day) to replace a leader. And by then, Labour might have shifted to a more pro-single-market position.

There is also, on Tuesday, a vote on a potentially far-reaching amendment concerning what would happen if – and after – the government were to be defeated in a “meaningful vote” on the final deal Theresa May brings back from Brussels this autumn.

This Lords amendment says parliament should then take control and instruct the executive through parliamentary motions how to proceed in further negotiations with Brussels. The government, seeing the huge constitutional significance of this, has put down its own counter-amendment to try to neuter its effect, saying that ministers would merely report back within 28 days.

Benn says this is the most important vote of all in the coming week – one that could “ensure that parliament gets to decide what should happen when we come to the end of the negotiations”.

After decades in which Eurosceptics have demanded curbs on the power of Brussels and a return to parliamentary sovereignty, this could, potentially, be a moment of supreme political irony – the moment when Remain MPs, if they muster the numbers, make that sovereignty a reality in order to stymie the ambitions of the Brexiters.

Key flashpoints

Customs union

This would prevent the European Communities Act being repealed until the government had spelled out steps it has taken to negotiate UK participation in a customs union. The Tories will oppose it, but Labour will vote for it, as will the SNP and Lib Dems. Around a dozen Tories would need to rebel for it to be passed. Anna Soubry and Ken Clarke are likely to rebel but may hold fire and vote for a stronger customs union amendment on the trade or customs bills in July.

European Economic Area

The amendment would force the government to make remaining in the EEA, and therefore in the single market, a negotiating objective. While there are more than a dozen Tory MPs who would have defied their government and voted for it, Labour has decided not to whip in favour. As a result it is unlikely to pass. But there could still be big Tory and Labour rebellions, showing strength of feeling in both parties for staying in the single market.

Meaningful vote

This says parliament must approve the withdrawal agreement. But it also says that if the deal the PM brings back from Brussels is rejected by MPs, parliament should effectively take control. The Tories have tabled a counter-amendment to try to neuter it, but the other parties will back it. There may be enough Tory rebels for this to be passed, but not definitely.

Environmental protections

After fears that EU environmental protections would be seriously weakened after Brexit, the Lords passed an amendment ensuring they would stay in place. With a few tweaks from former cabinet minister Oliver Letwin, the principle of keeping protections in place looks set to be accepted by the government. But campaigners are concerned that the amendment does not go far enough to ensure a green Brexit.