The president, whose praise for May follows his touting of her possible successors, met with the prime minister and corporate executives from the United States and United Kingdom as part of a day of negotiations on Trump's second day on British soil. The leaders' top priority is a possible bilateral trade deal to be negotiated once - or if - the U.K. leaves the European Union.
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May has been dogged by her failure to secure Brexit. She plans to resign Friday , days after Trump departs England, as head of the Conservative Party but remain as prime minister until her successor is chosen. It will be the new prime minister's responsibility to negotiate Brexit and a trade deal Trump wants for the U.S. and U.K. Trump has been sharply critical of May in the past but only had warm words for her Tuesday as he jokingly urged her to stay to "get this deal done."
"I think we'll have a very, very substantial trade deal," said Trump, extolling its virtues for both nations. "I think that this is something we both want to do ... we're going to get it done."
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After Trump suggested May stay on, most in the room chuckled. The two leaders later warmly chatted during a tour of 10 Downing St., the prime minister's office, as May pointed out a copy of the American Declaration of Independence. They also planned a joint news conference.
Traditionally, U.S. presidents avoid interjecting themselves in the domestic politics of other nations. But Trump is far from traditional.
Trump told the Sunday Times in an interview before arriving that Britain should "walk away" from talks and refuse to pay a 39 billion-pound ($49 billion) divorce bill if it doesn't get better terms from the EU.
That move, known as a "hard Brexit," could have a devastating impact on the U.K. economy, according to many experts, and stands in contrast to a previous White House position that the departure should be as painless as possible. Others in the U.K. are pressing for a second referendum that could keep the EU intact.
The president has also opined that Brexit party leader Nigel Farage, an outspoken advocate of leaving the EU without a deal, should be given a role in the negotiations. Farage, a divisive figure in Britain, has long been a Trump supporter. And while Trump has avoided criticizing May on this visit, unlike a year ago when he blistered her in a newspaper interview just before landing in London, the president has touted her rival, Conservative Party leadership candidate Boris Johnson, as an "excellent" leader for the U.K.
The economic meeting at St. James's Palace brought together 10 leading companies - five from the UK and five from the United States. CEOs and senior representatives from BAE Systems, GlaxoSmithKline, National Grid, Barclays, Reckitt Benckiser, JP Morgan, Lockheed Martin, Goldman Sachs International, Bechtel and Splunk were listed as attending.
While the business leaders gathered, protesters began to assemble across London, some of whom had the now-infamous Trump baby balloon bobbing in the air near Parliament Square. Leaders of Britain's main opposition party are due to join demonstrators at a rally in Trafalgar Square, just up the street from May's Downing Street office. Also in Trafalgar Square: a 16-foot robotic likeness of Trump seated on a golden toilet.
Feminists, environmentalists, peace activists, trade unionists and others demonstrated against the lavish royal welcome being given to a president they see as a danger to the world, chanting "Say it loud, say it clear, Donald Trump's not welcome here."
"I'm very cross he's here," said guitar teacher Sarah Greene, carrying a home-made sign reading "keep your grabby hands off our national treasures" under a picture of one of Queen Elizabeth II's corgis.
"I find him scary. My sign is flippant and doesn't say the things I'd really like to say."
One group came dressed in the red cloaks and bonnets of characters from Margaret Atwood's "The Handmaid's Tale," which is set in a dystopian, misogynist future America.
Demonstrators filled Trafalgar Square and spilled down Whitehall, a street lined with imposing government offices.
Many paused to photograph a robotic likeness of Trump sitting on a golden toilet, cell phone in hand. The robot caught the attention of passers-by with its recitation of catchphrases including "No collusion" and "You are fake news."
"It's 16 feet high, so it's as large as his ego," said Don Lessem from Philadelphia, who built the statue from foam over an iron frame and had it shipped by boat across the Atlantic.
Lessem, a dinosaur expert who makes models of prehistoric creatures, said "I'm interested in things that are big, not very intelligent and have lost their place in history."
"I wanted people here to know that people in America do not support Trump in the majority . and humor is my weapon," he said.
Leaders of Britain's main opposition party joined demonstrators later at the rally, just up the street from May's Downing St. office. Police have erected barricades to stop protesters marching past the gates of Downing St.
Labour Party leader Jeremy Corbyn, who declined an invitation to Monday's royal banquet for the president, was due to address the protest.
Emily Thornberry, Labour's foreign affairs spokeswoman, said Trump was "a sexual predator" and a racist who did not deserve the honor of a state visit hosted by Queen Elizabeth II.
Thornberry told the BBC that the leader of Britain's most important ally should be stood up to "the way you deal with a bully" because "if you bow down in front of them you just get kicked harder."
Not everyone in London was unwelcoming.
Lewis Metcalfe said he came to the city from his home in northern England to show support for the president.
"I'm obviously going to be a minority today," said Metcalfe, who wore a "Make America Great Again" cap. "I don't agree with all his policies. He's not the greatest president in the world, but he does get things done."
The U.S. president arrived in Britain at a precarious moment, amid a fresh round of impeachment fervor back home and uncertainty on this side of the Atlantic. The day of meetings with May follow a whirlwind of pomp, circumstance and protest for Trump, who had lunch with Queen Elizabeth II and tea with Prince Charles before a grand state dinner at Buckingham Palace.
The queen used her toast to emphasize the importance of international institutions created by Britain, the United States and other allies after World War II, a subtle rebuttal to Trump, a critic of NATO and the U.N.
But most of the talk and the colorful images were just what the White House wanted to showcase Trump as a statesman while, back home, the race to succeed him - and talk of impeaching him - heated up. Yet Trump, forever a counter-puncher, immediately roiled diplomatic docility by tearing into London Mayor Sadiq Khan.
As so often happens when Trump travels overseas, norms were shattered, including when the president complained about his television viewing options in the foreign capital and urged people to punish CNN by boycotting its parent company, AT&T.
Following Tuesday's focus on business and trade, Trump will use the next two days to mark the 75th anniversary of the June 6, 1944, D-Day landing, likely the last significant commemoration most veterans of the battle will see. The events will begin in Portsmouth, England, where the invasion was launched, and then move across the English Channel to France, where Allied forces began to recapture Western Europe from the Nazis.
French President Emmanuel Macron is expected to use the occasion to call for strengthening multinational ties the U.S. president has frayed.