The mood was sad, the music solemn. On Tuesday Rauf Denktash, the Turkish Cypriot leader who died last week, made his final journey through the streets of old Nicosia. A guard of honour and then a gun carriage bore him along. A white and red crescent flag was draped over his coffin. It symbolises the Turkish Republic of Northern Cyprus, the state – or illegal separatist entity, depending on your point of view – for which Denktash so stubbornly fought.

Hundreds of people turned out. They clapped as his coffin passed. Some stood on chairs to get a better view. "He was our Churchill. He was our Margaret Thatcher," said one mourner, 29-year-old Hasan Ayhan. "None of the other leaders measure up to him." The procession moved in slow-step from the president's palace – the former office of Cyprus's British governor, with heraldic lions at the entrance – through the shady alleys of divided Nicosia.

Denktash was a British-educated barrister, known for his quicksilver mind and twinkling humour. His support for Turkish Cypriot independence was so unyielding, so the joke went, that during his entire legal career he took only one brief. The decades-long leader of the Turkish Cypriots, he declared his own state in the north of the Mediterranean island in 1983.

Rauf Denktash. Photograph: Tarik Tinazay/AFP/Getty Images

Almost 30 years on, only Turkey recognises it. The Turkish government attended Denktash's funeral en masse, a sign of Ankara's continuing support for his project in the face of international rejection. The rest of the world supports the southern (Greek Cypriot) Republic of Cyprus. An EU member since 2004, the republic has been battered by the chilly crisis blowing in from the eurozone and Greece. Its credit rating was reduced this week to junk.

At the funeral, Denktash's grandson Rauf Alp Dentkash said his grandfather had left behind a "fully fledged state".

"It may not have recognition today. But it's democratic," he said. "The Turkish Cypriots are ready for a solution to the Cyprus problem so long as they are guaranteed their security. My grandfather was very much afraid that what happened to the Turks in Crete would happen here."

State or no state, the northern side of the island – which includes the laid-back harbour town of Kyrenia and the strategic port of Famagusta – survives only because of Turkish largesse. That and a garrison of 21,000-40,000 Turkish soldiers (the figure, like everything here, is disputed). The Turkish Cypriot republic remains economically isolated. There is an international embargo. There are no direct flights.

The prospects for a settlement, meanwhile, remain as distant as ever.

On Sunday the Greek Cypriot leader, Demetris Christofias, will meet his Turkish Cypriot counterpart, Dervis Eroglu, for the latest round of talks. These are taking place under the auspices of the UN secretary general, Ban Ki-moon, and the venue is Greentree, a cosy UN country retreat with log fires, a couple of hours' drive from New York.

The two sides met here in November, but neither expresses much hope that a deal is near. "I am very pessimistic," said Cyprus's foreign minister, Erato Kozakou Marcoullis. "It will take a miracle." The UN is exasperated at the failure of both parties to bring the talks to any kind of conclusion after decades of on-off negotiations and a 2004 referendum on reunification, which was accepted by the north but voted down by the south.

The UN wants a solution before Cyprus takes over the EU rotating presidency on 1 July. Turkey has signalled that it will suspend its links with the EU's presidency while (Greek) Cyprus sits in the chair.

The discovery of up to 8tn tonnes of gas off Cyprus's eastern coast is another UN headache. The Turkish Cypriots lay claim to a share of it. Rebuffed, they have started their own undersea explorations.

"No one ever lost any money betting on a non-solution to the Cyprus problem," said Hubert Faustmann, associate professor in European studies at Nicosia University. He said negotiations had "largely stalemated". They had got bogged down in disputes over governance – how a reunified island should be run, "cross-voting" – and what rights the minority Turkish Cypriot population should enjoy in a liberal democracy, he said.

In theory, both sides share a vision for the island's future. The new state would be a "bi-zonal" and "bi-communal" federation, with "one citizenship" and "one international personality". The problem is that no one agrees what this actually means. There is also a profound lack of mutual trust, with both leaderships accusing each other of obduracy, unreasonableness and a bone-headed refusal to compromise.

The lack of consensus spills over into the terms used to describe Cyprus's fraught modern history, encompassing British imperial rule, smouldering communal tensions and the Turkish military's invasion of 1974, which led to today's bitter partition. The Greek Cypriots want the right of return for ethnic Greek refugees (or their descendants), turfed out of their homes in the north and forced to flee. It is unlikely, though, that many would go back even if they could.

Turkish Cypriots reject the word invasion. They prefer the term "intervention". They point out that their community suffered too and was the victim of Greek Cypriot violence and ghettoisation from 1963 onwards. There is no agreement either on how to describe the tens of thousands of mainland Turks whom Denktash invited to the north since 1974, deliberately changing the island's ethnic balance. Are they settlers or immigrants? And do they have rights?

With talks going nowhere, a group of anti-globalisation protesters have taken an initiative. They are squatting on the green line running across Nicosia. Denktash opened a crossing here in 2003, allowing Cypriots from both sides to revisit their ancestral villages. The south allows Turkish Cypriots to cross but not Turkish settlers/immigrants. Tourists are free to stroll to and fro.

These days there is modest pedestrian traffic in both directions. On the Turkish Cypriot side there is a passport control booth, where you get an instant visa stamped on a piece of paper. Turkish Cypriots cross to visit Debenhams or Starbucks in Ledra Street; Greek Cypriots go the other way to gamble in the north's flashy casinos.

"The answer to the problem is to abolish capitalism. We need to get rid of elites on both sides," said Orestis, a 19-year-old Turkish Cypriot protester whittling a juggling stick with a large knife. After capitalism's fall, then what? "I like the idea of anarcho-syndicalism," he said. On a board nearby someone had scrawled: "I don't want to show my ID card to buy Efes," a Turkish beer. Others had planted trees in pots to "green" the green line.

Actually, Orestis may have a point. Both governments are made up of older, grey-haired men in their late sixties. They have childhood memories of dispossession and exodus. Some younger Cypriots believe that an agreement will only be reached once the current generation is gone, but others caution that the youngsters are more hardline than their older peers. Writing in the Cyprus Sunday Mail, the columnist Loucas Charalambous said Cypriots were sick and tired of the subject. "The Cyprob will follow me to my grave," he moaned.

Ahead of next week's talks, likely to be the last for some months, there are lots of accusations. George Iacovou, chief aide to Christofias, says the Turkish Cypriots have "retrogressed" from commitments made by Mehmet Ali Talat, the north's social democrat former president. Talat lost power in 2010. His successor Eroglu is an unreconstructed hardliner, Iacovou claims. Additionally, the Denktash team have failed to give any census figures for the north's population. Turkish Cypriots are now a minority in their homeland, outnumbered by mainland settlers.

The Greek Cypriots charge that the other side is, in reality, working on a secret plan B. Under this scenario, once talks fail and in a blame-game scenario, North Nicosia will push for an "upgrade" of its status. The economic embargo will end, with some international recognition and enhanced EU ties. Turkey can only accede to the EU when it recognises the Republic of Cyprus. But for the moment this is a far-off possibility, with Ankara cooling on EU union in the face of French and German hostility and economic meltdown.

This week Osman Ertug, the Turkish Cypriot chief negotiator, denied there was any plan B. "We only have plan A," he insisted. He said that any settlement of the Cyprus issue could take place only on the basis of political equality between the two communities, with any new future state "comprising two constituent states of equal status". He defended the huge number of mainland Turks now living on the island. "Immigration is an international phenomenon," he said.

Reaching back into the past, he observed: "There's been a community of Greek and Turkish Cypriots here for centuries. They lived in peace in Ottoman times. It was in British times things turned sour because of the British policy of divide and rule." He added: "British Romantics loved Greece. I love these poets myself. But love for ancient Greece shouldn't translate into the politics."

One suspects that Denktash would be pleased with the way the history he tried so hard to shape has worked out. His state, or non-state, has survived him and is still very much a reality.