JERUSALEM (Reuters) - Israeli government splits over the war in Gaza broke open this week, after the Defence Minister publicly demanded a clear strategy from Prime Minister Benjamin Netanyahu as troops returned to battle Hamas fighters in areas thought to have been cleared months ago.

The comments from Defence Minister Yoav Gallant, who said he would not agree to setting up a military government in the enclave, reflect growing unease in the security establishment at the lack of direction from Netanyahu over who will be left to run Gaza when the fighting stops.

They also brought out the sharp split between the two centrist former army generals in the cabinet, Benny Gantz and Gadi Eisenkot, who both backed Gallant's call, and the hard right nationalist religious parties led by Finance Minister Bezalel Smotrich and Internal Security Minister Itamar Ben-Gvir, who condemned the comments.

"That's no way to run a war," the right-wing Israel Today tabloid headlined its Thursday edition over a photo of Netanyahu and Gallant facing in different directions.

Apart from dismantling Hamas and returning some 130 hostages still held by the Islamist movement, Netanyahu has not articulated any clear strategic goal for the end of the campaign, which has killed some 35,000 Palestinians and left Israel increasingly isolated internationally.

However, backed by Ben-Gvir and Smotrich, both close to the West Bank settler movement, he has rejected any involvement in running postwar Gaza by the Palestinian Authority, set up under the Oslo interim peace accords three decades ago and generally seen internationally as the most legitimate Palestinian governing body.

Netanyahu, struggling to hold his increasingly fractious coalition together, has so far stuck to his pledge of total victory over Hamas. Afterwards, Gaza could be run by a "non-Hamas civilian administration with an Israeli military responsibility, overall military responsibility", he said in an interview with CNBC television on Wednesday.

Israeli officials have said that Palestinian clan leaders or other civil society figures may be recruited to fill the void but there has been no evidence that any such leaders, able or willing to replace Hamas, have been identified and no friendly Arab countries have stepped forward to help.

"From Israel the options are either they end the war, and they withdraw, or they establish for all intents and purposes a military government there, and they control the entire territory for who knows how long, because once they leave an area, Hamas will reappear," said Yossi Mekelberg, an associate fellow with the Middle East and North Africa Programme at Chatham House.

GUERRILLA TACTICS

Gallant's refusal to contemplate any form of permanent military government reflects the material and political costs of an operation that could stretch the military and the economy painfully, reviving memories of Israel's years-long occupation of southern Lebanon after the 1982 war.

Taking full control of Gaza would require perhaps four divisions, or around 50,000 troops, said Michael Milshtein, a former intelligence officer and one of Israel's leading specialists on Hamas.

While thousands of Hamas fighters have been killed in the campaign and Israeli commanders say most of the movement's organised battalions have been broken down, smaller groups have popped up in areas the army left in the early stages of the war.

"They are a very flexible organisation and they can adjust very quickly," Milshtein said. "They have adopted new patterns of guerrilla warfare."

The likely cost to Israel of a prolonged insurgency was illustrated on Wednesday, when five Israeli soldiers were killed by an Israeli tank in a so-called "friendly fire" incident, as Israeli troops fought fierce battles in the Jabalia area north of Gaza City.

Israel's military spokesman, Rear Admiral Daniel Hagari, said the military's job was to "break down those places where Hamas is returning and trying to reassemble itself" but he said any question of an alternative government to Hamas would be a matter at the political level.

Although most surveys show Israelis still broadly support the war, that support has been slipping, with more and more prioritising a return of the hostages over destroying Hamas. Such incidents may erode support further if they continue.

A taste of the broader social divisions likely to be unleashed has been seen in the long-running dispute over conscripting ultra Orthodox Torah students into the military, a move backed by Gantz and his allies as well as by many secular Israelis but fiercely resisted by the religious parties.

Netanyahu has so far managed to avoid a walk-out by either side that could potentially bring his government down.

But Gallant, who has already led a revolt against Netanyahu from within the cabinet over plans to cut the power of judges last year, has clashed repeatedly with Smotrich and Ben-Gvir and his latest challenge to the prime minister may not be his last.

(Additional reporting by Andrew MacAskill in London; Editing by Nick Macfie)

By James Mackenzie