With less than two months to go until Brexit, Jeremy Corbyn and his shadow cabinet gathered in the House of Commons at 10am on 5 February for their weekly meeting. Apart from a brief mention by the party leader of his unfruitful talks a few days earlier with Theresa May in Downing Street, there was no discussion of Labour’s approach to Britain’s impending departure from the EU. “We didn’t really talk about Brexit,” said one member of the party’s top team who was present.

The decision not to debate the party’s approach to the biggest crisis to have engulfed British and European politics in decades may have come as a relief to some shadow cabinet members. Emily Thornberry, the shadow foreign secretary, is said to have been reluctant to say much about Brexit in the last few meetings for fear that her views might leak to the press and fuel an impression of internal division. “It is not a great situation where the shadow cabinet cannot discuss the national crisis facing us all because they are all so divided,” said another source who was there.

Those at Labour’s top table, and the rest of the party, did not have to wait long, however, before details of the latest strategy that had been hatched inside Corbyn’s tight inner circle did emerge. The next day the leader’s aides briefed selected newspapers that Corbyn had written to the prime minister spelling out the conditions under which the official opposition would be prepared to back her Brexit deal. These included many of the same demands he had put to her the previous week – including a permanent and comprehensive UK-wide customs union.

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But there were significant changes and shifts of emphasis that made it look, to some, like Corbyn was warming to the idea of a Brexit compromise. The party’s oft-repeated insistence that any deal must deliver the “exact same benefits” of the EU single market was gone. And there was no mention of Labour’s commitment to keeping a second referendum on the table, the policy most dear to many activists and that the mass membership had succeeded in forcing Corbyn to accept after a gargantuan struggle at Labour’s annual conference in Liverpool last September.

News of the letter was welcomed by some Labour MPs in Leave seats and also some former Remainers who now believe the best way forward is an acceptable version of a soft Brexit. But in other sections of the party there was uproar.

The former leadership contender Owen Smith said he was so incensed at the suggestion that Labour might be preparing to “row in behind” a modified Conservative Brexit deal that he was thinking of leaving the party.

The frontbencher Clive Lewis, from the left, responded the next day by saying he would never vote for Brexit and wanted the leadership to stick to its commitment on a people’s vote. And leftwing activists, many of whom had joined Labour because Corbyn had promised them a say over policy, were outraged too. One told the Observer that it was not just the shadow cabinet which had been bypassed: “The question this raises above all is: where is the role of members in this?” Where was the democracy idealistic young pro-European activists had been promised by Corbyn?

Michael Chessum, a former member of Momentum’s steering committee, who now campaigns for a second referendum, said that if Labour MPs backed Brexit because of deals with the Tories, it would be shameful, and he suggested that local members should deselect them. “Disagreeing about stopping Brexit altogether is one thing,” he tweeted. “Voting through, or allowing through, a Tory deal is quite another. It’s just inexcusable, and MPs who do so will, quite rightly, be held accountable by members.”

For much of the past week the leadership has been engulfed in a series of internal controversies, and not only on Brexit. This weekend, it performed a sudden U-turn in yet another row over antisemitism, in which Luciana Berger, a Jewish MP who is eight months pregnant, was threatened with deselection by Corbyn-backing members of her local party in Liverpool.

Facebook Twitter Pinterest Luciana Berger, a Corbyn critic, was facing a no-confidence vote from her local party until it was suddenly withdrawn. Photograph: Yui Mok/PA

A no-confidence motion in Berger was withdrawn following pressure from an evidently panicked leader’s office. Labour’s general secretary Jennie Formby also bowed to pressure and agreed to publish details of hundreds of antisemitism complaints against party members and progress to date in dealing with them, having batted away MPs’ calls to do so last Monday in a move that caused widespread dismay.

As it firefights rows on many fronts there is a stubbornness about Corbyn’s leadership on issues such as antisemitism and Venezuela, and a deliberate vagueness on Brexit, that irks many Labour MPs and increasingly members of the party. But, according to very senior figures, there is also no doubt that there is concern at the highest level about rising levels of discontent and talk of breakaways. As reported by the Observer last weekend, a group of at least six Labour MPs, all persistent critics of Corbyn, are planning to resign the whip and form a new breakaway party within weeks. Moves are also being considered by a far bigger group of “mainstream” MPs to do the same, after Brexit.

Supporters of the Labour leader have piled on to social media to say good riddance to the splitters. But the discontent is no longer confined to what Corbyn loyalists like to dismiss as a Blairite rump of MPs. Unrest and disillusion have spread to sections of the pro-EU left of the party, and the activist base. Party membership has fallen rapidly. Labour is dropping in the polls. Two surveys have put the party seven points behind the Tories in the past week and Corbyn’s personal ratings have nosedived, particularly over his handling of Brexit, to record lows. At a time when discontent with politicians is arguably greater than ever, debate at Westminster now focuses not only on whether the Tories can survive Brexit as a coherent fighting force, but whether Labour can hold itself together too. As one shadow cabinet member put it: “Things are already tense. I think some of our MPs will split off pretty soon. But if we are seen to allow a Tory Brexit there is no telling what will happen. All hell will break loose.”

After Corbyn sent his five-point letter to May on Wednesday, and Labour Remainers reacted with fury, the leadership and the small circle who had been involved in drafting it were quick in their attempts to pour cold water on the flames. The Brexit spokesman Keir Starmer made clear that the letter had been signed off by him, in consultation with the leader’s office, and was really just a restatement of the party’s existing position. Starmer and the shadow chancellor, John McDonnell, went public to insist that a second referendum was still “on the table” and the fact that it was not mentioned meant nothing. In fact nothing had changed. The commitment to the “exact same benefits” of the single market was also implicit in the five points.

But, while some Labour MPs were supportive of the initiative, including Lucy Powell and Stephen Kinnock (hardly full-throated Corbyn backers), doubts remained in the minds of many others who are committed to fighting May’s Brexit tooth and nail. Wes Streeting, the Labour MP for Ilford North, told the Observer it was time for clarity from the leadership because the country was heading for a catastrophe which had to be avoided.

“It really is decision time for Jeremy Corbyn now on whether he actually backs a people’s vote or whether he really wants to facilitate Brexit. No ifs, no buts, no tables. We have to be clear.”

Facebook Twitter Pinterest Jeremy Corbyn poses for a selfie after a speech at the Labour local government conference in Coventry. Photograph: Aaron Chown/PA

Writing today on guardian.co.uk, Alena Ivanova, a Momentum and Labour activist, says there is a danger that the leadership’s so-called “constructive ambiguity” on Brexit, and reluctance to involve the wider party in policy, could drive many Corbyn supporters away. While the Corbyn letter might have been designed to split the Tories, it could also split and fracture the Corbyn support base. “It may have the inadvertent effect of disillusioning Labour supporters further who might interpret this as an offer to co-deliver what was always a Tory-driven project,” she writes. “Labour should stand firm on its commitment to vote down May’s deal. All MPs should be effectively whipped to do so. It is the only principled thing to do – it offers a chance to defeat the government, to get the general election we need now, not in 2022, and to then go back to the public, honestly stating that Labour never wanted or participated in the Tory Brexit dream, and that it has a different, positive, socialist vision for our relationship with the EU. Then, we let the voters decide.”

A strongly pro-Remain member of the shadow cabinet said there were two interpretations of what Corbyn was up to. “The charitable view is that he is doing this so when May cannot accept his demands he can then move to support a second referendum, saying it is the option left and he has tried everything. But the second view is that Corbyn wants to get Brexit over the line because that is what be believes, which in my view would be disastrous.”

On Saturday, in a speech in Coventry, Corbyn adopted a familiar formula on Brexit that left few people any the wiser as to where Labour will end up or where he really stands. He said his plan – as outlined in the letter to May and involving a permanent customs union of the kind the prime minister refuses to accept – could “win the support of parliament and bring the country together”. If the Tories could not support Labour’s demand he said – unequivocally, and decisively – there should be a general election, even though Labour recently lost a confidence vote that could have delivered one. Its main route to an election has already been cut away.

Corbyn then gave a typically cautious nod to the idea of a second referendum, saying that without an election “we will keep all options on the table – as agreed in our conference motion – including the option of a public vote”.

After a week in which Labour MPs and members from all wings of the party demanded clearer leadership, and talk of a breakaway group grew ever louder, it was hardly the kind of address to reassure the doubters. “This is more kicking the can down the road,” said one disillusioned MP. “The longer this goes on and the closer Brexit comes the worse it gets. We should be clearly demanding another referendum by now because that is our policy. But it is clear he cannot bring himself to do so. If the leader cannot implement his party’s policy there is no hope for us.”