Members of the newly formed "Independent Group" sit together in the British Parliament.Reuters

LONDON—Among the things that unite the British lawmakers who resigned from the Labour and Conservative parties this week, perhaps the greatest commonality is their shared support for Britain having a second Brexit referendum.

Paradoxically, their departures might have made that goal much harder to achieve.

The walkouts began on Monday with seven Labour MPs, who announced in a press conference their decision to resign en masse and sit as a new group of independents, citing their frustration with the party’s lack of leadership on Brexit and its failure to tackle anti-Semitism within its ranks. By Wednesday, the “gang of seven” became 11, as another Labour lawmaker and three Conservatives joined the so-called Independent Group. Not since the founding of the Social Democratic Party here in the 1980s has either party experienced so many defections.

And yet, when British lawmakers gathered in the House of Commons on Wednesday for their weekly questions to the prime minister, everyone avoided the elephant in the room: that Labour and the Tories have lost nearly a dozen lawmakers between them at a time when—with Brexit only weeks away—they arguably need them the most.

But what the fledgling group of lawmakers lacked in any formal acknowledgment from either the Conservative or Labour leadership, they made up for in size. Though not a formal party, in a matter of days they have become the fourth-largest grouping in Parliament, larger than the Northern Irish Democratic Unionist Party (which props up Prime Minister Theresa May’s Conservative government) and equal to that of the Liberal Democrats.

Few things unite this group of Labour and Conservative defectors. Their views run the gamut, from when to end austerity to whether to nationalize the railways. But on their reasons for leaving their parties, they are somewhat in agreement. For one, both sides are frustrated by what they see as a hard-line takeover of their former parties, from the far-left loyalists of Labour Party leader Jeremy Corbyn to the outsize influence of the hard-line pro-Brexit lawmakers within the Conservatives. All 11 argue that it is not their values, but rather those of their parties, that have changed.

In terms of hard policy, one thing has brought them together. They have all advocated for a so-called People’s Vote—another Brexit referendum. If their united goal is to persuade Parliament to back a second vote, though, their defections might have instead helped achieve the opposite.

For another vote to happen, it needs the parliamentary backing of at least one of the major parties, something neither Labour nor Conservative leaders have officially stated they are willing to do. And with MPs such as Labour’s Chuka Umunna and the Conservatives’ Anna Soubry—vocal supporters of a second referendum who have now defected—no longer affiliated with those parties, the likelihood of either party changing its stance seems to have only diminished further.

For now, the People’s Vote campaign does not appear to see it that way. In a statement responding to the defections on Wednesday, it said that while it respected the reasons put forward by the 11 lawmakers to resign, it would continue to work with MPs from both parties to secure support for a second referendum. “We are not a political party,” a People’s Vote spokesperson said, “nor are we ever going to allow ourselves to be associated with just one faction of any political party.”

What is similarly unclear is whether advocating for a second referendum will prove enough to unite the new movement.  “I suppose everyone is looking at this thinking, It can’t be as unplanned as it looks,” Anand Menon, the director of the London-based U.K. in a Changing Europe research institute, told me. “But the thing is, maybe it is.”

In Brexit Britain, that’s par for the course.

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