Prime Minister Rishi Sunak of Britain faces a litany of woes, from a double-digit deficit in the polls to a grinding cost-of-living crisis. But this week, his biggest source of agita comes from the “five families,” a loose coalition of right-wing factions in his Conservative Party that is threatening, yet again, to torpedo his asylum policy.

That these Tory potential rebels would style themselves after the five organized crime families that once ruled over the rackets in New York City attests to both the enduring appeal of mob movies like “The Godfather,” and the lawmakers’ own self-image as political tough guys. It’s also more than a little double-edged: Most of the leaders of the real five families wound up dead or in prison.

Still, the five-families label has stuck, and this week, the rebels are back with a sequel to their noisy campaign last month to force Mr. Sunak to harden legislation that would deport to Rwanda asylum seekers who arrived on the British coast in small boats. As in December, there have been late-night meetings, hastily arranged media briefings, defiant social-media posts, and offers to Mr. Sunak that he can’t refuse.

When it’s all over, with a vote in Parliament likely on Wednesday evening, the prime minister is expected, perhaps narrowly, to keep his Rwanda policy alive.

But the recurring drama reveals a Conservative Party splintered into multiple rivalrous factions, with some lawmakers seemingly more intent on plotting their own futures than on uniting the party for a coming election against the opposition Labour Party.