As Israel’s Prime Minister Netanyahu calls for early elections, it’s difficult not to notice that the date he has chosen for voting—January 22nd, 2013—is quite close to the U.S. inauguration. But the relevant date is not January 20th, it is November 7th, 2012. One feels that only once we know who the U.S. president will be into 2017 can the Israeli campaign truly begin. It has been said that Netanyahu could have an effect on the U.S. elections. But the reverse is true as well.

Before going further, let’s establish that the U.S. presidential contest won’t affect who will get the most votes in Israel’s elections. That is foreordained: it’s Bibi. Polls show him with a commanding lead; Haaretz, for example, has him dominating even his strongest potential challenger, Kadima’s Tzipi Livni, who probably won’t run. (The one wild card is former Prime Minister Ehud Olmert, who is in between corruption trials yet still popular.) Put another way: Netanyahu would not call early elections if he weren’t going to win them.

But what is unknown is how the other parties will do and which ones Netanyahu will seek to form a coalition with following re-election—an essential question, given that no party, not even the hyper-dominant Labor Party of the 1950s, has ever won the majority of Knesset seats by itself. And this is where the U.S. election could affect things.

The current Israeli coalition is, by the admission of Netanyahu’s own top aide, the most right-wing in history. Aside from the Independence Party, which is basically Defense Minister Ehud Barak’s one-man shop, it features Likud, the traditional right-of-center party, a good deal of whose members are to the right of Bibi; Yisrael Beiteinu, the revanchist party popular with Russian immigrants; and a smattering of religious parties that, particularly on issues domestic and settlement-related, make Netanyahu look like a bleeding heart. (Iran does not obey these configurations—Netanyahu and Barak are the hawks, while the rabbis reportedly still need convincing.)

Flash to America for a second. The relations between the Israeli and American leaders are the worst they have been in at least two decades, arguably longer. No matter how much blame you apportion to Obama, you need to save plenty for Bibi, who in the past few months has made matters worse by all but openly campaigning for Mitt Romney.

If Romney wins, Bibi’s in the catbird seat, and his bet on Romney, inapproriate though it was, may pay off. But if Obama does, then the prime minister has a problem. Many Israelis lack warm feelings for Obama, but many more understand the importance of their country’s special relationship with America. Thus, they were duly freaked out by the openness with which Netanyahu seemed to be rooting for one side, given the possibility that the other side might win; thus, it was correctly perceived as a domestic political move when Barak met with Chicago Mayor Rahm Emanuel, in effect an Obama administration official, to assure him of continued Israeli friendship, and did so behind Netanyahu’s back.

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In sum, four more years of Obama means four more years of Netanyahu needing to get along with a guy none too fond of him, who, having achieved re-election, will have less reason to need to get along with him. The upshot is that an Obama victory will make it more difficult for Bibi to form another far-right-wing government. Instead, as the Wall Street Journal reports, he will be under pressure to join with parties of the center—Barak’s, sure (if his party gets the two percent needed to qualify!), but also Kadima, whoever is running it, and perhaps also the new party created by television personality Yair Lapid. The personal animosity between Obama and Netanyahu would remain to some extent, but a centrist coalition would be more palatable to a second Obama administration, because they would be more like-minded on many issues of both substance, particularly the settlements and the Palestinians, and temperament. (U.S. cabinet ministers tend to have more in common with the politicians from the middle-of-the-road Kadima than from the Sephardic ultra-Orthodox Shas; and while she will not be his next secretary of state, Hillary Clinton was angered by ultra-Orthodox protests in favor of female modesty earlier this year, and the views of those protesters are represented in the current government.)

And a centrist coalition could have profound ramifactions for Israel. It will create a more hospitable climate for a law mandating the drafting of more religious (and even Arab) citizens into the military or civil service; for economic policies meant to address the inequality that drove Israelis to the streets in the summer of 2011; and even, maybe, for a softer line on the Palestinian question. (Again, Iran is a different story. Any government run by Netanyahu and including Barak is likely to maintain the stances of the current one.)

Before creating his current government, Netanyahu offered to form a coalition with Livni’s Kadima, which actually received a few more Knesset seats in the 2009 elections. (By contrast, the unity government Netanyahu formed for a few months this year, when he welcomed Kadima in, still included the right-wing parties, substantially diluting the centrists’ influence.) Forming a center-right government in response to an Obama re-election would be entirely in keeping with his track record of flexibility in the pursuit of self-interest.

And finally it is worth remembering that the U.S. election will not be the prime driver of Israeli elections—their politics are local, too. But it won’t be nothing. If a vote for Romney is in some small way a vote for the Israeli status quo, then a vote for Obama is a vote for encouraging a more liberal Israeli society.