As the prospect of a non-Bibi government begins to emerge in Israel, some in Washington, Europe and Turtle Bay cheer, hoping for a sharp left turn in Jerusalem. Hey there: as Larry David warns, curb your enthusiasm.

If agreements are finalized, the center-right coalition built on a promise to unseat Benjamin Netanyahu, the longest serving prime minister in Israel’s history, will, at least for its first two years, be headed by Naftali Bennett.

Mr. Bennett said today he’d gather top members of his Yamina party on Sunday to get their nod before finalizing a political agreement. If they give their okay, the new government would be announced early in the week, just before the expiration of a 31-day mandate forming a governing coalition.

That mandate was granted by President Rivlin to Yair Lapid, the head of Yesh Atid (“there’s a future”), the Knesset’s second largest party after Mr. Netanyahu’s Likud. Earlier Mr. Netanyahu was given a similar mandate but failed to convince 61 Knesset members to support a government he’d head.

Mr. Lapid’s agreement with Mr. Bennett, according to which the latter would serve as prime minister for two years, now seems to have paved the way for Mr. Netanyahu’s unseating.

The Haifa-born Mr. Bennett, whose parents hail from San Francisco, is a long-time champion of West Bank settlement policies and an advocate of hardline military stance on Gaza and Lebanon. His party’s name, Hebrew for “to the right,” is a good indication of where he’d like to lead the country as prime minister.

Will his coalition partners allow it?

Mr. Lapid, a harsh opponent of Mr. Netanyahu from the center-right, is reportedly expected to at first serve as foreign minister and alternate prime minister. After two years (if the coalition lasts that long) he would switch roles with Mr. Bennett and become premier.

Other coalition partners would include the leftist Labor and even further left Meretz parties. Arab parties, including Communists and Islamists, would support the coalition but remain outside the government.

So yes, left and center-left politicians will hold key portfolios, but the top positions in the new government are expected to be held by people affiliated with the right, including many who have cut their political teeth alongside or under Mr. Netanyahu.

Benny Gantz, currently Defense Minister in Mr. Netanyahu’s government, was responsible for overseeing the most-recent Gaza war. In the new coalition he is slated to continue leading Hakirya, Israel’s equivalent of the Pentagon. A former army chief of staff, Mr. Gantz is likely to maintain a tough line against Hamas and other aggressors.

Avigdor Liberman, once widely seen as a political architect of Mr. Netanyahu’s successful ascent to the premiership, has since become a bitter foe. He’s slated to become finance minister, guaranteeing the economy will mostly remain as free-market as it has been in recent decades.

Gideon Saar, until recently a top Likud member, is considered a behind the scenes Knesset operator. Now also an anti-Bibi voice, Mr. Saar would be the new government’s justice minister. He wouldn’t easily cave in to international pressure on legal battles in Jerusalem and the West Bank, or so goes the theory.

Ayelet Shaked, Mr. Bennett’s partner in the Yamina party and likely to become minister of internal affairs, reportedly remains hesitant to join a coalition with Leftist partners or rely on Arab parties’s support. Yet the betting is that she’d end up following Mr. Bennett’s cue after all.

How long would such a coalition last? If the recent past is any indication, power sharing among unnatural partners can easily collapse for the smallest of reasons. Renewed hostilities in Gaza, for example, might well force Arab partners to bolt, leaving the coalition with no Knesset majority.

Yet Messrs. Bennett and Lapid bank on the idea that a top source for division among Israelis is Mr. Netanyahu’s personality. The prime minister is as loved by parts of the electorate as he is despised by others, who point to the numerous court cases against him. Take Bibi out of the equation, their reasoning goes, and you can unite the Israelis.

Add to that the external pressures: a large majority of Israelis sees the attacks from the far left wing of the Democratic Party, or from much of Western Europe and others, as unfair, not to mention counterfactual. And then there’s Iran and its proxies.

“I’d advise all Zionists to go back and repurchase the houses they sold in Europe, the U.S. and elsewhere before those houses become more expensive than they are today,” the commander of Iran’s Revolutionary Guard Corps said Saturday in a speech supporting Hamas.

Western politicians ignoring Israel’s real enemies and Iran-led terrorists with genocidal intent might well become the glue that will repair Israel’s broken politics. Or not. Conversely, they could pave the way for a Bibi comeback.

Twitter @bennyavni.

Correction: Yesh Atid is the party that is headed by Yair Lapid; the name of the party was incorrectly given in the bulldog edition.