TEL AVIV—Since taking over in June as Israel’s prime minister, Naftali Bennett has tried to construct a new political order. The biggest obstacle might be the man he displaced, his predecessor, Benjamin Netanyahu.

Mr. Bennett, a right-wing politician who will be in Washington this week to meet President Joe Biden, heads an unstable coalition of leftist, right-wing and Arab parties and by necessity must govern from the center.

He has tried to call every minister each week to ask how they are, according to coalition members. His tweets add the handles of other ministers to congratulate them on the government’s

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