This past summer, out of view of the press and the spotlight, an unlikely cabal of secular and religious politicians began plotting to shake up the Israeli chief rabbinate. The conspirators: Avigdor Lieberman’s ultra-nationalist Yisrael Beiteinu party, the ultra-Orthodox Shas party, and a renegade rabbinic organization called Tzohar—three of the strangest bedfellows in Israeli politics. Their plan, if successful, would break the ultra-Orthodox stranglehold on the country’s rabbinate and install a moderate religious Zionist chief rabbi for the first time in decades.

“Avigdor Lieberman was very interested in promoting Tzohar,” said one person with close knowledge of the proposed deal, “to make sure that they...