* Once an obscure academic, Conte wins public support
* Clash with Salvini strengthens his reputation
* International endorsements help domestic image
ROME, Sept 5 (Reuters) - U.S. President Donald Trump and European Council President Donald Tusk have sharply different political views, but agree on one thing: admiration for Italian Prime Minister Giuseppe Conte.
"A very talented man," Trump waxed on Twitter last week, rooting for Conte as a political crisis threatened his future.
"He is one of the best examples of loyalty, responsibility, and also has a sense of humour," enthused Tusk.
The fact such disparate characters see eye-to-eye on Conte highlights one of the premier's strongest attributes - his ability to build bridges between discordant political worlds and win over both populist and establishment spirits.
On Thursday he took the helm of a new government giving him an opportunity to heal rifts with the European Union that soured his last coalition, strengthen the stagnant economy, and cement his own role as a decisive political leader.
It is a tall order for a man with no background in politics before being plucked from the obscurity of academia last year to head up an administration comprising the anti-system 5-Star Movement and far-right League.
That coalition collapsed last month when the League walked out, hoping to trigger early elections. But rather than cave in to demands he resign, Conte unexpectedly held his ground, giving 5-Star time to work out an unlikely new pact with former fierce rivals the centre-left Democratic Party (PD).
His stout resistance in the face of heavy pressure from League leader Matteo Salvini won him the admiration of a previously hostile PD, enabling him to stay in his job.
"Salvini the savage was tamed by a fierce yet gentle bullfighter," left-leaning Repubblica newspaper wrote approvingly of Conte, a man it had previously dismissed as an "ectoplasm", "a puppet" and "a politician by proxy".
Conte, 55, a law professor from Florence who is not affiliated to any party, was seen by the League and 5-Star as the ideal frontman for their coalition. They wanted someone who they could oversee and who would not steal their limelight.
That is who they got - at least initially.
In his first parliamentary appearance, Conte asked permission from 5-Star leader Luigi Di Maio to say something to the house. "No," came the curt response, caught on camera.
But his confidence has grown, especially abroad where he travelled alone, giving him a greater degree of autonomy.
He twice managed to head off the threat of EU disciplinary action against Italy and established good relations with senior European leaders, who were highly suspicious of Salvini.
"This helped him build his stature at home," said Sergio Fabbrini, professor of political science and director of the school of government at Luiss university in Rome.
Extremely dapper, with a handkerchief permanently poking out of his breast pocket, Conte's popularity has grown as voters warmed to his measured tones, which stood in stark contrast to the fire and fury of his ever-bickering deputies.
A poll published on Monday said he had an approval rating of 51%, second only in Italy to that of the president. By contrast, Salvini had a rating of 36% and Di Maio 26%.
His support had climbed six points in a month, boosted by a powerful speech in parliament on Aug. 20 when he mauled Salvini, accusing him of looking to "tip the country into a spiral of political uncertainty and financial instability".
A religious man, Conte also attacked Salvini's habit of brandishing the rosary in public - a criticism that stung.
"You cannot say something like that in Italy without authorisation from the bishops and cardinals," said Fabbrini, making clear that Conte must have won the backing of Italy's powerful, hidden establishment to take on Salvini.
"But he cannot govern with that establishment. He needs to be his own leader," he added.
(Reporting by Crispian Balmer; Editing by Andrew Cawthorne)