By Erika Solomon and Tarek Amara

TUNIS (Reuters) - More than any threat of military force, the power of Tunisia's main trade union may be what pushes the Islamist-led government to accept opposition demands for it to quit.

Inspired by the army-backed removal of Egypt's Islamist president, the secular opposition in Tunisia has taken to the streets to demand a new government. Thousands of its supporters have been joined by ordinary Tunisians fed up with rising instability and economic stagnation.

All this had seemed to leave the ruling Ennahda party unmoved - until Tuesday. Then the Tunisian General Labor Union, courted for days by the opposition, came out in support of creating a new technocrat government. Ennahda, a moderate Islamist party, later said it was willing to consider that plan.

Whereas in Egypt military power decided the Islamist government's fate, in Tunisia the economic muscle of the 600,000 member union - known by its French acronym UGTT - may prove decisive; just one day of strikes can cost the country hundreds of millions of dollars.

"It is the force capable of influencing the street and its leaders can topple the government," said opposition activist Sofian Chourabi. "The UGTT can reshuffle the political cards because of its manpower and its political and economic weight. It can play the role that the Tunisian army can't."

The UGTT, which represents workers across the economy in both the state and private sectors, has been a major political player since it staged regional strikes in 2011. These helped protesters to force out autocratic ruler Zine el-Abidine Ben Ali, setting off a wave of uprisings across the region.

More recently economists estimate a one-day strike, called by the UGTT to protest against the assassination of leftist politician Mohamed Brahmi, cost up to $422 million last week.

Economist Moez Joudi told local newspaper Assabah that the stoppage caused a stock market dive and pushed the Tunisian dinar to its lowest value ever against the dollar and euro.

Such influence gives the union a forceful hand to play in a country suffering economic stagnation and rising unemployment - problems which are already increasing frustration with the government.

"A MESS OF DIVISIONS"

A leftist workers' body, the UGTT is ideologically already close to the secular opposition that has been on the offensive since the Egyptian army intervened during anti-government protests to remove president Mohamed Mursi.

Brahmi's killing, the second of a Tunisian opposition politician to be blamed on militant Islamists this year, drew further support to the opposition's cause.

The Tunisian army may have played a role in Ben Ali's overthrow by refusing to shoot demonstrators. But unlike the Egyptian military, which also helped protesters to topple fellow autocrat Hosni Mubarak in 2011, it remains politically weak.

It has few strings to pull and, unlike the Egyptian military, little economic privilege to protect.

"The Tunisian army is neutral and won't force its will... It has no established custom of playing a political role," said a source close to the military.

Though most opposition leaders publicly deny wanting military intervention, average supporters make no secret of their desire for an "Egypt scenario".

Young activists have even copied the Egyptian Tamarud (Rebel!) campaign group by organizing their own petition calling for the government to quit. Their movement (also called Tamarud)says it has collected more than 200,000 signatures.

But Ennahda, elected with 42 percent of the vote in October 2011, remains a force to be reckoned with.

In a face-off on the streets, it is unclear whether the opposition could force Ennahda to accept its demand to dissolve not only the government, but also Tunisia's transitional Constituent Assembly - which is weeks away from finishing a draft constitution.

Some observers suggest internal security forces could play a more influential role. Under Ben Ali, Tunisia was a police state where Interior Ministry forces had influence and power, but they are now as divided as the general population.

At recent protests in Tunis, Reuters reporters heard some security force members arguing about whether to fire tear gas at demonstrators.

"The Interior Ministry is a mess of divisions. Some departments are now in the hands of Ennahda, others are holdovers from the former regime," said Tunisian analyst Youssef Welati. "I don't think they have any kind of decisive role. And neither does the army. The most they can do is decide not to suppress protests."

A NATIONAL UNIFIER?

Nevertheless, sources close to the opposition say leaders trying to form a rival "salvation government" are proposing security force members, such as Rachid Ammar, the former army chief, and former defense minister Abd Elkarim al-Zbidi.

But the institution the opposition is best poised to benefit from is the UGTT. Historically, the unions have been a powerful force and the UGTT organized against French colonial authorities before independence in the 1950s.

"We are a national organization whose role it is to rescue the country," Sami Tahri, the assistant secretary general, told reporters on Monday.

The union has hinted it may consider more strikes if the political situation doesn't improve but is also trying to create for itself a more unifying role. It has rejected opposition calls to dissolve the constitutional assembly, apparently aware of complaints that this would be destructive and could lengthen Tunisia's transition to democracy.

That stand seems to have eased Ennahda's fears and allow it to open up to the possibility of a new government.

"The UGTT can find a deal that guarantees the continuity of the state but at the same time meets some of the demands of angry protesters," activist Chourabi said. "It may be the one who can create a consensual exit plan from the crisis."

(Editing by William Maclean and David Stamp)