Ms. May’s Brexit was sunk primarily by pro-leave politicians in her Conservative Party and its coalition partners, who claim there is a better deal to be had despite repeated and unequivocal statements to the contrary by E.U. leaders. Parliament will now vote on the option of leaving the union on March 29 without an accord, which could cause severe disruption on both sides of the English Channel. But that extreme course, favored by Conservative hard-liners, has already been rejected once.

The only measure Parliament may be able to agree upon is requesting an extension of the exit deadline from Brussels. That would avert the disaster of a no-deal Brexit for now, but the political crisis could then be prolonged for months more, with no clearer path toward a solution.

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The impasse results from a broad failure of political leadership. Ms. May might have been able to put together a centrist majority behind a Brexit formula that minimized the break with Brussels, had she been willing to spurn the Conservative Party hard-liners. But she never tried. Labour leader Jeremy Corbyn, for his part, made no effort to find a solution but instead has angled to bring down Ms. May’s government and force a general election. That gambit, too, already failed once but could now have a better chance of success.

A general election might produce a government that had the backing to push through a Brexit accord, either on Ms. May’s terms or a softer version. Another option, favored by many in and outside Parliament, would be a second referendum. Now that the true costs and consequences of leaving the European Union are clearer, the result might be different; if it were the same, Parliament would have a mandate to enact Ms. May’s accord or something like it.

For the embattled prime minister and other responsible actors, the central challenge in any election or further parliamentary debate is overcoming the demagoguery of those on the Conservative right and Mr. Corbyn’s left who insist, against all evidence, that Britain can make a clean break with the continent while simultaneously preserving its economic health and the fragile peace in Northern Ireland, which depends on an open border with the Irish republic. The refusal by Conservative ideologues and left-wing opportunists to abandon those unsustainable positions explains most of what has gone wrong in British politics during the past several years.