* Aden chaos worsened by absent police, rule of gunmen

* Aden is strategic trade, shipping hub

* Islamists share coalition's enmity towards Houthis

* Islamist activists decry gender mixing

By Sami Aboudi

DUBAI, Oct 27 (Reuters) - In Aden, Yemen's most cosmopolitan city, the masked Islamist gunmen seem to have burst onto the streets from some other world.

They storm university classrooms to demand men and women to stop studying together, charge into supermarkets firing in the air to force the female cashiers to cover up, harass families celebrating a Muslim holiday on the beach.

Three weeks ago, four suicide bombers detonated car bombs at a temporary Yemeni government headquarters and two Arab coalition outposts.

With that, Islamic State served notice that it was bringing its bloody campaign for a Sunni Muslim caliphate to a city where separatist fighters, Arab armies and an embattled rump government had just forced out Shi'ite militiamen. For many residents, it now seems like a baffling conflict is about to enter its darkest phase.

"Aden is at risk of falling to Islamists and if it does, that would be a disaster for everybody," said a prominent Yemeni academic who, fearful of the consequences of speaking out, asked not to be identified.

For months, Aden has been pre-occupied with repelling the Houthis, a movement of fighters from northern Yemen's Zaydi branch of Shi'ite Islam who emerged as the country's most powerful force last year after allying themselves with former president Ali Abdullah Saleh and seizing the capital Sanaa.

Three months ago, local fighters and a Saudi-led alliance of Arab states recaptured Aden from the Houthis and brought President Abd-Rabbu Mansour Hadi's government back to the city from exile in Saudi Arabia.

Hadi's Arab-backed government announced plans to reestablish its authority in Aden before a military push to recapture the capital. But instead of imposing strong authority over the port, it has watched as the city has slid further into chaos.

Contingents from Saudi Arabia, the United Arab Emirates and Bahrain have been reinforced by Sudanese troops, deployed around the airport while fighters loyal to Hadi are trying prevent gunmen from publicly carrying arms.

They have been unable to stop gunmen appearing from nowhere to enforce their vision of Islamic laws, stage religious trials and conduct drive-by shootings.

Residents fear the disorder may pave the way for Islamists to take control of the strategic shipping and trade hub at the bottom of the Arabian Peninsula.

"It would certainly not look good for the coalition to liberate Aden from the Houthis and Saleh and then leave it to fall to al Qaeda and Islamic State," said the academic.

DRONE WAR

The Islamists are hardly new to Yemen: for years, Yemen's rural tribal areas have sheltered one of the most powerful regional branches of al Qaeda, which the United States has targeted in a secret drone war, in alliance first with Saleh's government and then with Hadi's.

But what is new is their assertiveness on the streets of Aden, which served as the capital of an independent Communist South Yemen until 1990.

Aden has grown more conservative since North and South Yemen were unified into a single state. Its only brewery was shut two decades ago and few women now go out unveiled. Nevertheless, residents still regard the port city, set around a spectacular mountain peninsula, as far more cosmopolitan than the surrounding countryside or the northern highland capital Sanaa.

Since the Houthis captured Sanaa a year ago, al Qaeda fighters have exploited chaos and expanded their influence across southern and eastern Yemen.

Fighters seized heavy weapons from army camps, organised tribal resistance to the Houthis and filled the security vacuum as Hadi's government was beaten back.

For the past several months, al Qaeda has effectively ruled over parts of Yemen's vast eastern Hadramout province, including its provincial capital, the strategic port of Mukalla.

In Lahej and Abyan, the two southern provinces adjacent to Aden, mediators last week failed to persuade militants to hand over to government forces 55 armoured vehicles and tanks and 22 rocket launchers seized from the military.

With Hadi's government now trying to fight its way back into power, it has avoided direct confrontation with al Qaeda, which in turn has avoided attacks on government targets. Many Yemenis believe Hadi and his Arab allies reached some sort of truce with the militants.

Al Qaeda's strict Sunni Muslim ideology makes it natural enemies of the Shi'ite Houthis, and the Houthis say Hadi's government and al Qaeda have become de facto allies on the battlefield.

But although al Qaeda is still probably the biggest jihadist faction in Yemen, it is no longer the only one, and the emergence of Islamic State this year has brought a force that has made clear it will observe no truce.

The world's most violent jihadist group, Islamic State considers all Shi'ites apostates deserving of death. It served notice it had become a serious presence in Yemen by staging a quadruple suicide bombing of Houthi-linked mosques in Sanaa in March, killing nearly 150 people and wounding hundreds more, the deadliest attack of its kind in Yemen's history.

On Oct. 6 in Aden it staged its biggest attack against Hadi's government and his Arab allies with another quadruple suicide bombing that killed at least 15 people including four Emirati soldiers.

"Daesh has a dogma," said Ibrahim Freihat, a senior analyst at the Doha branch of thinktank Brookings. "It can attack all parties at the same time, and one cannot expect to see logic in what it does."

GUNS AND NO JOBS

Since Hadi and his government returned to Aden in September, the security situation in the city has continued to deteriorate, causing resentment among residents desperate for stability.

Gunmen - whether loyal to al Qaeda, Islamic State or some other group, or acting on their own - have capitalised on frustration with the lack of jobs or public services and the absence of government authority to swell their ranks.

"You have a young population with guns and with no jobs. What do you expect to get?" said Yemeni analyst Abdel-Bari Taher. "It is a dangerous mix."

Hadi's government says it is taking steps to get armed youths off the streets, including plans to incorporate into its forces some 5,000 members of the so-called Southern Resistance, part of a separatist group that helped drive the Houthis out of Aden.

But the southern fighters, who fought on Hadi's side in a tactical alliance against the Houthis, worry that their movement risks being overshadowed by Islamists, unless the government does more to assert its authority.

"If the government fails, al Qaeda will rise up and control everything, like in Mukalla," Mohammed al-Saadi, a Southern Resistance leader, told Reuters earlier this month in the ornate lobby of the seaside al-Qasr hotel where Hadi had set up a temporary administration. Just days later, the hotel became one of the targets of the Oct. 6 Islamic State attacks.

Masked men now appear in groups of four or five in coffee shops, universities or at the beach. In the latest incident on Sunday, residents said gunmen stormed a supermarket, firing shots in the air demanding female cashiers cover their faces.

A lecturer at the University of Aden said five masked men had stormed into classrooms to demand that male and female students be separated.

"The time of debauchery and fornication is over," he quoted one of them as saying. "This is the time to abide by Islamic Sharia laws."

A student, who asked that her name be withheld, said she was with a group celebrating last month's Eid al-Adha holiday at the beach when men showed up to lecture them that music was banned by Islam.

President Hadi's press secretary, Mokhtar al-Rahbi, said the government was "aware of the fragility of the security situation in Aden."

"The (Oct. 6) incident accelerates the need to put the security situation in the province in order." (Additional reporting by Angus McDowall, editing by William Maclean and Peter Graff)