Diplomatic relations between Turkey and Britain continued to prosper on a hot day in Istanbul yesterday, despite a regal run-in with a couple of politically sensitive chess sets.

The Queen and the Duke of Edinburgh were joined by David Miliband, the Foreign Secretary, on day three of their state visit as they were invited to sample wares from the Grand Bazaar on the banks of the Bosphorus.

The Queen calmly negotiated her way through the displays of carpets and hookah pipes, leaving her husband to pause by the novelty chess sets.

First he studied one featuring Osama bin Laden (Tony Blair was king, Donald Rumsfeld his queen; George Bush and Saddam Hussein were merely pawns) Then Prince Philip picked up an English knight from a Crusader-themed historical chess set. “Always the losers,” the Prince said, referring to the British with his customary tact.

The Foreign Secretary had less luck with the ice-cream stall. The minister was fooled by a particularly cheeky vendor who invited him to take a vanilla cone from the end of a long spatula, before snatching it from his grasp. Repeatedly. The trick prompted a line from Mr Miliband, heard all too uncommonly from a Cabinet Minister: “If I stick my tongue out, will you let me lick it?” he asked, hopefully.

The Bosphorus may be no more than 17 nautical miles long – a mere thread on the map of Turkey, compared with the Tigris and Euphrates to the east but it is also the crucial river that separates Europe from Asia.

In this location heavy with symbolism, the royal couple shared a boat trip with President Gül and his wife Hayrunnisa across the glittering water where East meets West.

They left the stalls outside Kabatas school, where pupils have been working together on a science project with children from Haggerston School and Hackney Community College, East London, and boarded the boat bound for Istanbul Modern, Turkey’s first modern art museum.

Once a warehouse on the wrong side of the river, the museum opened in 2004 – only days before the European summit in Brussels invited Turkey to start talks on EU membership. The trip to the gallery, where the Queen met the British architect Zaha Hadid, is intended as yet another example of Turkey’s readiness to join the European Union, itself a running theme of this royal visit. This week Turkish newspapers have featured pictures of the Queen on their front pages and coverage has been largely positive.

“In politics and diplomacy there is no such thing as coincidence,” said Didem Vardar, a journalist with Turkish state television, referring to the Foreign and Commonwealth Office sending of both the Prince of Wales and the Queen to Turkey within the past six months.

“I think the attention that the UK is giving Turkey with all these visits can be seen as a sign of the potential support from the UK in the future.

In the afternoon, the Duke was treated to another trip along the river in a search and rescue boat by the Turkish Straits Vessel Traffic Service, which has been policing this strategic waterway since 2003.

Supertankers squeeze dangerous chemical cargoes through a city of more than 15 million people every week. About 140,000 million metric tonnes of oil passed by Istanbul last year. As new oil and natural gasfields develop around the Caspian Sea, pipe-line politics will increase Turkey’s leverage on the international stage.

Finally, a cocktail reception aboard HMS Illustrious brought to an end Turkey’s three-day flirtation with Old Europe. The Turkish President and his wife were piped on board. Once the gin and tonics had been drunk, Beat Retreat and Ceremonial Sunset were played. Then it was farewell to the Turkish hosts and adieu to an aspiring European nation.

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