A watermelon slice of bright sunlight falls across my breakfast table, bouncing off the gleaming silver cutlery, around the perfect white shell filled with delicate tropical flowers and off the bubbles glistening on the side of the Perrier bottle. Preston, my 6ft-tall dreadlocked butler, moves swiftly to adjust the huge cream umbrella that is shading me. I continue tucking into a plate of melon, berries and kiwi fruit, while Rexford, my chef, delivers a basket of freshly baked pains au chocolat.

A few feet ahead of me, the Caribbean Sea is glinting, its blue the truest I have ever seen – though the view behind is almost as good. Three storeys and 8,755 sq ft of stark white modernist architecture rear up, a composition of triangles, squares and circles that mirror the shape of the sailboats in the bay beyond the strip of white beach.

For just £10m, it could be mine. The Brazilian Emerald, one of a trio of geometric villas at the Altamer resort on Shoal Bay West, Anguilla, is up for sale. It has five enormous bedrooms, a crescent-shaped balcony with ocean views, seven marble bathrooms and a gym, as well as a gallery of South American artefacts. It is hard to imagine a more luxurious – or exclusive – holiday home.

Tiny Anguilla, the most northerly of the Leeward Islands, has 33 beaches that are considered among the best in the Caribbean, but it is to the almost mile-long crescent of pristine sand at Shoal Bay West that celebs have traditionally flocked. Denzel Washington threw his 50th-birthday bash here; Liam Neeson and Natasha Richardson have sunbathed. Jay-Z and Beyoncé frolicked on a boat in the bay, no doubt later sailing round the eastern cove past the houses rented by Robert De Niro and Halle Berry. And, I discover, to my shameful glee, it was after holidaying here in the Brazilian Emerald that Jen and Brad broke up in 2005.

The driving force behind this resort of super-villas is Rebecca Eggleton, 43, a driven, linen-clad American who trained as an auditor. After moving to Moscow in 1993, when her husband, Michael, got a job there with Trust Bank, she was keen to have a house somewhere warm. “We went all over the Caribbean, but, after coming to Anguilla three times, we knew this was where we wanted to build,” she says. “The people are so genuine, and it is one of the few places in the world where you can actually relax.”

After admiring Covecastles, a Le Corbusier-style 1980s resort of 15 white steel-and-glass castles in the sand, further down the same beach, the Eggletons commissioned Myron Goldfinger, its New York-based architect, to build a home for them “at a cost of millions and millions”. His design for a five-bedroom, 7,790 sq ft house, with a stunning walkway projecting from the second floor over the swimming pool below, became reality in 2000, in a villa now known as the Russian Amethyst. Goldfinger’s interior-designer wife, June, was employed to decorate it with furnishings and paintings from Russia and Turkey.

“It was so fabulous – we knew we didn’t want anything unattractive built around us, so we started to acquire the neighbouring land,” Eggleton recalls. “I knew then that this could be a business. The Caribbean desperately needed state-of-the-art, high-tech villas with all the mod cons.”

By 2004, three sawtoothed, futuristic villas had been completed: the third, the African Sapphire, was decked out with Ghanaian kente cloth and bead chandeliers commissioned from women’s refuges in Johannesburg. These “private museum residences”, each sleeping at least 10, have acres of hurricane-proof glass, a private cinema room and a fitness centre, and have been rented out for between £20,000 and £45,000 a week.

Eggleton is not content to stop there. She has commissioned Goldfinger to create designs for another six homes on Shoal Bay West, for sale for up to £32m each, that will make it one of the most exclusive strips of beach in the world. The architect has gone several steps further with these new designs, geared towards the squillion-aires’ market. Although they will be white modernist palaces like the original three, they will be even grander. With anything from 16,000 sq ft to 25,000 sq ft of interior space, they will have six or seven bedrooms and innovative features such as elevated walkways, dramatic glass lifts, coral-rock waterfalls, aquariums and roof gardens. Each will also have its own tennis court and separate guesthouse.

“It is an artistic achievement to have so many superstructures that are all unique but relate to one another,” Goldfinger says of the project. “It is a mural, a composition. A cove of art.”

Eggleton would like the owners of the nine villas to keep up the gemstone theme, with matching decor – suggested names include Australian Opal, Indian Garnet, Thai Pearl and, for the largest and most expensive, Siberian Diamond. She is also keen for buyers to place the properties in the Altamer rental scheme, through which owners receive 60% of the booking fee. Chef, butler and gardeners can be arranged.

“We’re searching for an international group of high-net-worth individuals,” Eggleton says. “We see it as a networking opportunity: billionaires from, say, Turkey, Russia and the USA. And we will be flexible with the new designs, building them as they are sold to suit people and their professions: if it’s an investment banker, we’ll build them a trading floor; if it’s a musician, a recording studio.”

Eggleton will also ensure that her buyers have easy access to their Caribbean home. In case the island’s airport gets too congested with Gulfstream Vs, she is building a 101-slip megayacht marina right next to the resort. Working with Island Global Yachting, which develops luxury marinas, she will put 30 slips up for sale for between £50,000 and £4m each, depending on length. There will also be bou-tique shopping, a first for the sleepy island.

The super-rich are, of course, a nervy breed, and privacy is paramount (although the average Anguillian laughs at the security forces people bring to the island). “Villa orientation means you will never be able to see your neighbour,” Goldfinger says.

So, which villa do I want, and do I want June to decorate it, I muse over a lunch of duck and watermelon salad, washed down with the resort’s sommelier-chosen house white, a rather fetching Chilean sauvignon blanc.

My last day at Altamer dawns, and there’s just time for an in-suite massage with Louis “Hands” Price, the resort’s recommended masseur, who is flown by clients to Mustique or Kensington for bodywork. As he kneads my glutes, he exhorts me to breathe in the sunshine – but it doesn’t allay my sense of dread. Goodbye, Preston and Rexy. Goodbye, white sails on endless blue vistas and melt-in-the-mouth chocolate terrine.

“You are leaving the villa this morning, aren’t you?” Eggleton asks politely, clearly concerned by my query the night before about squatters’ rights in Anguilla. Okay, okay, I’m going. But if I ever get hold of £10m – and a megayacht – I’ll be back.

The International Gems of Altamer are for sale through Savills International; 020 7016 3740, www.savills.co.uk/abroad, www.altamer.com

Treasure islands

ANGUILLA

Temenos, a Baccarat development of 116 homes in Merrywing Bay, has a golf course designed by Greg Norman. Three five-bed oceanfront homes, with private pool, yoga tower and hot tub, are for sale for between £3.94m and £7m through HMI Properties; 01932 580790, www.hmiproperties.co.uk

ST LUCIA

On 19 acres of beachfront at Rodney Bay, the Landings development has one-bedroom marina-view flats with shared pool for sale from £285,000; 0845 217 7851, www.thelandingsstlucia.com

ST BARTS

In fashionable St Jean, this four-bedroom villa has an infinity pool and bay views. For sale for £5.5m with Quintessentially Estates; 0845 224 3658, www.quintessentiallyestates.com

BARBADOS

Golden Eye, a plantation-style house near Sugar Hill, on the smart west coast, has four bedrooms. For sale for £1.96m with Knight Frank; 020 7629 8171, www.knightfrank.com

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