A beaming David Cameron strode into the White House from the South Lawn on Wednesday knowing that he had just secured the prize cherished by every British prime minister.

In the warm Washington sunshine, Barack Obama had, moments earlier, delivered a gushing homage to the wonders of the Anglo-American special relationship. "We meet to reaffirm one of the greatest alliances the world has ever known," the president said, in remarks that would have done Ronald Reagan proud.

The Camerons knew a high five would be out of place. So Sam Cam, who was described by the newspaper USA Today as a "fetching British aristocrat", proudly patted her husband on the bottom as they entered the White House.

It was one of the lighter moments in a formal welcoming ceremony as Obama pushed protocol to the limits to give the prime minister the biggest welcome accorded any world leader this year.

Obama rolled out the red carpet, literally and politically. US military forces, with crewcut hair and severe looks from central casting, lined up to be inspected by the president and his guest. A 19-gun salute echoed around the South Lawn shortly after the Camerons arrived in a black US government limo, as US military bands played the British national anthem.

Once that part of the pomp and ceremony was over, Obama went into political overdrive. Standing alongside Cameron on a podium, he said: "Through the grand sweep of history, through all its twists and turns, there is one constant: the rock-solid alliance behind the US and the UK. The reason is simple. We stand together and we work together and we bleed together and we fall together in good times and bad, because when we feel our nations are secure, our people are more prosperous, the world is a safer and better and more just place."

The prime minister reminded Obama that two of their grandfathers fought in the "greatest ever British and American operation in history" – the retaking of France in 1944. Cameron said: "Whether it is defeating the Nazis, standing up to the Soviets, defending the Korean peninsula or hunting down al-Qaida in Afghanistan, there can be no more tangible illustration of our two nations defending our values and advancing our interests than the mutual sacrifice made by our servicemen and women."

Cameron and Obama acknowledged that the Anglo-American relationship has not always been sweetness and light. The president joked that it was exactly 200 years since the British arrived in Washington "under somewhat different circumstances", when they burned down the White House. Cameron feigned embarrassment at the behaviour of the British. "I am a little embarrassed, as I stand here, to think that 200 years ago my ancestors tried to burn this place down," he said. "I can see you've got the place a little better defended today. You're clearly not taking any risks with the Brits this time."

The ringing endorsement of the special relationship by Obama may have come as a surprise to some who had thought his priority was to rebalance US interests away from the Atlantic and towards the Pacific. But one British source said the combined efforts of British and American forces in Afghanistan and the danger posed to the US by a collapsing eurozone meant that was never going to last long. "The forces of gravity propel us back together again," the source said.

From the warm reception on the South Lawn, Cameron and Obama headed off to the Oval Office for a one-on-one meeting where they discussed international issues. They were then joined by their most senior cabinet ministers – William Hague, George Osborne and their counterparts, Hillary Clinton and Tim Geithner – as they assessed the Nato mission in Afghanistan, the future of Syria's president, Bashar al-Assad, and the Iranian nuclear threat.

After their talks, the two leaders both pledged at a press conference in the Rose Garden to press ahead with the "draw down" of troops in Afghanistan. This will see Afghan forces take the "lead combat role" by the middle of next year before the withdrawal of all Nato combat troops by the end of 2014. "We are now in the final phases of our military mission," the prime minister said.

Obama, who said the next stage in the transition to Afghan control would be decided at the Nato summit, said he was not surprised that voters overwhelmingly want the mission to end immediately. "Why is it that poll numbers indicate people are interested in ending the war in Afghanistan? It is because we have been there for 10 years. People get weary. They have friends and neighbours who have lost loved ones. No one wants war. Anybody who answers a poll question about war saying enthusiastically we want war probably hasn't been involved in a war."

In the traditions of the special relationship, which experiences bumps even in its most productive phases, there was the odd tricky issue. Cameron raised Britain's extradition treaty with the US, which recently saw a retired Kent businessman sent to the US. British and American officials are to examine the implementation of the treaty as Cameron presses for more British citizens to be tried at home.

The Camerons rounded off their trip with a state dinner, held in a marquee on the South Lawn. Over lunch at the state department, the prime minister had revealed his wife's excitement about one person on the guest list. "We were looking through the guest list last night in bed and Samantha said: 'The star from my favourite movie is going to be there.' I said: 'Is is Ben Kingsley from Gandhi or Peter O'Toole from Lawrence of Arabia?' No, it's Chevy Chase from Caddyshack."

Bison wellington?

On the "winter harvest-themed menu" at the White House:

First course Brussels sprouts, applewood smoked bacon

Second course Spring garden lettuces, shallot dressing, shaved breakfast radish, cucumbers and avocados

Main course Bison wellington, a red wine reduction, French beans, cipollini onions

Dessert Warm meyer lemon steamed pudding with Idaho huckleberry sauce and newtown pippin apples

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