* New UN-backed body acts to stamp authority in Tripoli

* Charting Libya's recovery along with cenbank, oil corp

* Faces struggle to absorb or disband powerful militias

* Success key to tackling Islamic State, raising oil output

By Aidan Lewis

TUNIS, April 4 (Reuters) - In its first days in Tripoli, Libya's U.N.-backed unity government has tried to impose its authority swiftly by ordering a freeze on ministry budgets and securing the prime minister's office with the help of a powerful armed brigade.

A violent reaction by hostile militias has so far not materialised as feared. And from a heavily guarded naval base, the new government's leaders have begun charting the sprawling North African state's economic recovery in conjunction with the central bank and the National Oil Corporation (NOC).

But they still face a daunting task as they strive to unify and rebuild fractured institutions, boost oil production, and disband or absorb armed brigades that have established power bases while drawing on public salaries.

Those steps will be crucial not only to the government's survival, but also to countering Islamic State militants that have seized some pockets of Libya and to rescuing an economy in free fall due to reduced oil output and crude prices.

The unity government's leadership, or Presidential Council, arrived in Libya last Wednesday without securing the formal approval of either of the country's rival parliaments based in Tripoli and in the east of the country.

It has relied instead on the support or acquiescence of armed brigades that have controlled the capital since 2014, and earlier backed the self-declared National Salvation government.

But that same support risks stiffening opposition to the unity cabinet in Libya's east, the base of the National Salvation government's political adversaries and powerful anti-Islamist military commander Khalifa Haftar.

They may fear a western coup by their armed, Islamist-leaning rivals in Tripoli and the powerful port city of Misrata.

Aguila Saleh, president of the eastern parliament, hinted at such concerns when he said on Saturday that statements by some members of the Presidential Council "suggest they are not comfortable with the military".

"We will not allow the unity government to be under the control of militias in Tripoli," he said.

FICKLE LOYALTIES

The brigades, who cast themselves as the guardians of the 2011 uprising that overthrew veteran autocrat Muammar Gaddafi, have proved a disruptive presence in the past, and their loyalties can be fickle.

Any attempt to integrate them into national security forces answering to the unity government, or to stop paying their salaries for defying any such step, could trigger violence.

"They are supporting the national unity government because they are very cleverly repositioning themselves in the new environment," said Riccardo Fabiani, a senior North Africa analyst at Eurasia Group.

"That doesn't mean they will support the government all the time. As soon as divisions appear in the unity government they will side with one minister or another, like they did until 2014."

As well as appointing Haftar as military commander in the east, the government there has tried to set up parallel structures including an NOC and a central bank.

These bodies are considered largely powerless, however, and the Tripoli-based NOC and central bank have continued to operate on a national level throughout Libya's crisis.

However, oil output has been slashed by labour disputes, a lack of security, and regional and ethnic power struggles, and the eastern NOC has persisted in efforts to sell oil separately.

Ibrahim Jathran, a leader of the paramilitary Petroleum Facilities Guard (PFG) previously aligned with the eastern government, last week pledged support for the unity government and promised to reopen oil ports that his forces have blockaded.

But in a sign of the potential for further splits, Jathran's move prompted Battalion 152, a rival faction within the PFG, to declare that its loyalties lay solely with Haftar.

The eastern government has said it opposes any transfer of power before the unity government secures a long-sought vote of approval from the eastern parliament. The vote is a requirement of the U.N.-mediated deal under which the unity government was created, and is seen as symbolically important.

But the eastern NOC said on Monday that, even if the vote took place, it would carry on working out of Benghazi, the largest city in eastern Libya.

Haftar, with backing from Egypt's military, could also continue to operate independently, said Fabiani.

"The formal transfer of power and recognition of the national unity government is not the biggest obstacle. The biggest obstacle is what actually happens on the ground."

(Writing by Aidan Lewis; Editing by Mark Heinrich)