At first, President Bush and British Prime Minister Tony Blair (search) strained to find common ground. Asked what interests they shared shortly after Bush took office, the best he could come up with was that they used the same brand of toothpaste, liked sports and loved their kids.

Nearly three years later, the two leaders enjoy an intense bond forged by their passionate defense of the need to invade Iraq.

From many angles, it seems like an improbable relationship: a conservative president and a liberal prime minister with distinctly different styles.

Blair is smooth and urbane, an Oxford-educated barrister with polished rhetoric. Bush, on the other hand, pines for escapes to his rural Texas ranch, confuses audiences with his mangled sentences and makes no secret of his mediocre record at Yale.

Even so, they profess a fierce loyalty to each other.

"He's a smart, capable, trustworthy friend," Bush said of Blair last week. "He's the kind of person with whom I like to consult, a person I'm proud to call friend, because he's willing to make the tough decision and stand by it."

The friendship, galvanized by Iraq, is all upside for Bush, giving him an articulate spokesman about the need for war, adding more than 7,400 British troops in Iraq and serving as a bridge to Europe when other allies refused to go along.

But it has come at a heavy cost to Blair. His approval ratings have tumbled. Hundreds of thousands of people took to the streets to protest a war in which more than 50 Britons have since died. Some 60,000 demonstrators are expected when Bush visits London this week, and the security level there has been heightened.

Blair has lost two government ministers and seen his support erode among fellow Labor Party (search) members in Parliament. A weapons adviser, David Kelly, apparently killed himself in July after a tempest over intelligence. Nearing the end of his most stressful year in office, Blair was hospitalized last month with heart palpitations.

After the pummeling Blair has taken for standing with Bush, many observers wonder why the prime minister remains so loyal to the president. Public satisfaction with Blair has slumped from 49 percent in April to 33 percent last month in a Guardian-ICM poll.

"There's no doubt that Blair's popularity has been hurt by the war, as has Bush's," said Michael Mandenbaum, author of a book on 21st century international relations, "The Ideas That Conquered the World."

"You have to wonder whether this one is just a marriage of convenience," said James Goldgeier, a Europe specialist at the Council on Foreign Relations (search). "It's hard to imagine they have a lot in common as individuals, and they have different political orientations."

There are also sharp disagreements between the governments they head.

Britain's trade secretary, Patricia Hewitt, said last year that she and Blair were "bitterly disappointed" by Bush's decision to slap tariffs on imported steel — duties that the World Trade Organization (search) recently declared illegal.

Many Britons are angry that nine of their countrymen are among those being held at the U.S. detention camp for suspected terrorists at Guantanamo Bay, Cuba (search).

And Blair gently chided Bush on global warming in July, urging him to demonstrate more leadership on the environment.

The dispute over the Kyoto global warming treaty was a rare public display of disagreement. Some critics at home have accused Blair of being a lap dog for Bush and not confronting him more publicly, more often.

Blair said he saw little point in that during a time of war.

"I don't believe it is very sensible when you are in a coalition and you are fighting a war and then fighting a peace in very difficult circumstances to be mouthing off every so often," he told American reporters.

In a British newspaper Sunday, Blair sought to demonstrate that he and Bush have their differences.

"Where Britain's national interests are best served by airing them publicly — as for instance over our different positions on global warming or steel tariffs — I don't hesitate to do it," Blair wrote in the News of the World.

The president, meanwhile, said "I'm really looking forward to (the trip). It's going to be a fantastic experience."

Aides to Bush say that despite their disparate world views, the two leaders have a strong personal chemistry, trading jokes during their frequent meetings and weekly phone conversations.

Some analysts say Blair sticks with Bush out of principle and a tactical calculation that strong ties are in Britain's interest.

"He believed that to be effective in a world in which America is the predominant power, that the only way to do this was to be close to the United States and try to have whatever influence on the policy he could by being close," Goldgeier said.

"The other option is the French approach: Lead a group and counter American policy. I don't think Blair wanted to put the (United Kingdom) in that position," Goldgeier said.

Michael Worcester, chairman of the British polling firm MORI, said Blair's popularity has suffered more from his perceived failure to deliver such government services as health care, education and transportation than it has from his alliance with Bush.

While the president's visit may seem to come at an inopportune time for Blair, administration officials and outside experts say he can easily withstand the fresh criticism that will come when they meet Thursday at No. 10 Downing Street (search).

Unlike Bush, Blair does not face election within a year; his current term ends in June 2006.