Take it from me, there is nothing new under the tropical sun, at least not when it comes to travel. But here’s an experience that feels as if it might not only be new but hot too, in every sense. I have just returned from a group of islands so remote and so pristine that there were times when I wondered if I should be telling anyone anything about them at all. There are no bars, no shops, no big hotels and no roads, unless you count the strips of concrete poured through the centre of some villages. Tourists arrive by the trickle. In ten days, I saw three yachts. And to begin my long journey home I had to wade ashore before catching the first of five flights back to London.

These are classic desert islands. Except they are not deserts. Deserted, yes, in that hardly any are inhabited, but not barren. They are green and tangled; jungly piles of luxuriant rain forest. And they are set in a sea so clear and so blue it could be cut up for necklaces, or used in the construction of kingfishers. I imagine the South Pacific was like this 60 years ago. Small numbers of people have been going to the islands for years: they are a grail for scuba divers. Sea Trek Sailing Adventures, the company running my cruise, is more than 20 years old. But, and this is the point, I could believe I was the first outsider to find them. These are the islands of Raja Ampat in eastern Indonesia, 12 minutes south of the equator, where the Indian and Pacific Oceans meet. There are more than 1,500 of them, clusters of gnarled limestone karst, almost ornamental in their beauty. Weaving between them in fast ribs (reinforced inflatable boats) was like touring God’s rockery.

The biggest have beaches, villages and cliffs that rise in pointy heaps of green bush dense as plumage; most are little more than sprouts of limestone topped with shocks of vegetation. Some – the really photogenic ones – are shaped like mushrooms, poking from the water on ‘stalks’ of rock eroded by the sea. The archipelago is not just spectacular scenically; it’s opulent with nature. Once described as a ‘species factory’, Raja Ampat contains around 70 per cent of all known corals – ten times the number found in the Caribbean – and more than a thousand types of tropical fish. If the islands are romantic, so was the boat from which I saw them. She’s a pinisi, a traditional two-masted sailing ship built by the Buginese people on the island of Sulawesi. Adapted in the 19th century from European schooners, they are still used for inter-island transport. Their construction, by hand and eye, has been designated a Masterpiece of Heritage by UNESCO.

My pinisi, the Ombak Putih, which means White Wave in Indonesian Bahasa, was launched in 1996 as a 12-cabin passenger ship. With a sharp prow, tall masts, dipping sheer line and seven prickly sails, she has enough attitude to fly the Jolly Roger. She’s a rugged, broad-shouldered boat, built of ironwood and teak, gaff-rigged, 42 metres long and latticed with rigging. Boarding was like stepping into the pages of R.M. Ballantyne, who wrote The Coral Island, an adventure story from my childhood, except on Ombak Putih, I was greeted with cold towels and fruit juice. Below deck, the cabins are two-thirds taken up by comfy king-sized double beds. They are abutted against bulkheads and the side of the ship, so the least incontinent of a couple sleeps on the inside.

With portholes set in great black planks of ironwood, you might be on a man o’ war, but for the little copper washbasins, air conditioning, en suite shower rooms and lighting operated by eight separate switches. We feasted outdoors under a large awning on buffets of fresh fruits and lavish salads and hot dishes such as chicken in coconut batter, jack fruit curry, steak, shrimp and fish – groupers and snappers speared by John, the steering mate. Soft drinks were included. Alcohol was priced not much above cost: about £1.60 for a beer; £17 for a bottle of wine, both local. It reinforced my feeling that Sea Trek is out to share the experience, not exploit it.

We boarded in Sarong, a scruffy port of freight, fishing and ferries, at the western end of West Papua, the Indonesian half of the island of New Guinea. Within four hours of sailing we were snorkelling. We snorkelled every day, swimming in water as warm as whispers. Currents wafted us across forests of coral; we saw fish by the thousand – angelfish, parrotfish, damselfish, triggerfish, clownfish and wrasse. There were blizzards of fish, like psychedelic snow, luminous in myriad hues of red, yellow, orange, gold, green, silver, purple and blue. Snorkelling is like birdwatching… if birdwatchers could fly. On one day we swam into a limestone cavern, beneath great drooling stalactites, and then in a lake of brackish water populated by stingless jellyfish. There were hundreds of them, benign and gelatinous, little lavender-coloured translucent discs, and others like toadstools, orange and pulsating, no bigger than fists. Imagine being nuzzled by a swarm of fried eggs: weird.

We had lectures on Alfred Russel Wallace, who, in the islands of Indonesia, arrived at the same conclusions about natural selection as Charles Darwin, his contemporary, reached in the Galapagos. We landed at villages and saw hornbills and sulphur-crested cockatoos. On Arborek Island child dancers greeted us on the jetty. They had painted faces and headdresses made from cassowary feathers. A sign read: ‘Please be quite [sic] we have worship.’ At Lobo we visited the primary school and donated a water filter along with picture books warning of the perils of plastic. My expedition was bookended by two extraordinary encounters. At 4.30am on the first morning a group of us waded ashore on Waigeo, the biggest island of Raja Ampat. On the beach, in the dark, I changed into walking boots under the bow of a beharu fishing canoe. Dogs barked and a cock crowed as we tramped through the sleeping village. Using a machete, the local guide cut walking poles from saplings beside the path.

We climbed for nearly two hours. Around us echoed the whoops, warbles and whistles of the waking forest. Then, clear and fluting, came a different call: tiou, tiou, tiou. ‘A bird of paradise,’ breathed my guide. We came to a rudimentary canvas hide. On the ground, 30 metres in front of us, was a Wilson’s bird of paradise, the smallest and arguably the most spectacular of all 39 varieties. It was scratching in the earth and gave the occasional, inexplicable neurotic skip. Its colours were lustrous. What designer would dare marry sky blue and iridescent green with dazzling yellow and scarlet, let alone add a tail of two dainty curlicues, more like scrolls of fine ironwork than feather?

On the last day we swam within touching distance of a whale shark, the most docile and massive fish in the sea. This one, about six metres long, broad-backed, Armani grey and mottled with big white polka dots, was half the size to which they can grow. It was still impressively big. It was in constant graceful motion, its eyes, unblinking, aware and trusting of the dozen or so swimmers around it. But its most awesome feature was its mouth, almost the width of its huge, spatula-shaped head: think Julia Roberts smiling. Every now and then it would gape open, like the loading door of a car ferry, to ingest a maelstrom of water and nutrients. I feared if I came too close I would end up like Jonah. As we left, the whale shark raised its great head from the water as if in farewell. It was a memorable finale to an unforgettable trip.