Donald Trump's visit puts Brexit Britain's dependence on show

* U.S. President Trump visits Britain for first time

* Britain seeking to shore up trade ties as Brexit nears

* Anti-Trump protests expected in London

* NATO summit, Putin meeting underline diplomatic difficulties

By Guy Faulconbridge and William James

LONDON, July 5 (Reuters) - When Donald Trump visits Britain next week, Prime Minister Theresa May will have to face a harsh reality: Brexit makes Britain more dependent than ever on an alliance with the most unpredictable U.S. president in living memory.

Sandwiched between a NATO meeting and a summit with Russia's Vladimir Putin, Trump's first visit to Britain as president comes at one of the most important junctures for Europe and the West since the 1991 fall of the Soviet Union.

From challenging Western assumptions about the EU and free trade to courting the Kremlin and North Korea's leader, Trump has delivered on his promise of an "unpredictable" U.S. foreign policy.

That leaves May, who held hands with Trump at the White House during her visit after his inauguration, in a difficult position as she seeks closer trade ties with the United States to offset the disruption of leaving the EU on March 29, 2019.

"The irony is that by leaving the EU, the United Kingdom will be less useful to Washington as an ally but it will also need the United States much more," said Jeffrey A. Stacey, a former State Department official in Obama's administration.

"So May has been thrown into the arms of the most unpredictable U.S. president in living memory," Stacey said.

Over 50,000 people have signed up for a protest on Trafalgar Square in central London against the Trump visit, which will include a meeting with Queen Elizabeth and possibly even a round of golf at his Turnberry course in Scotland.

Even taking account of Trump's penchant for deal making, the visit is likely to be heavy on rhetoric about an increasingly lopsided "special relationship" and short on specifics such as the details of a post-Brexit trade deal.

For supporters, Trump and Brexit offer the prospect of breaking free from what they see as obsolete institutions and rules that have weakened the United States and its allies relative to competitors such as China.

But for many British diplomats, Brexit marks the collapse of a 70-year British strategy of trying to balance European integration with a U.S. alliance based on blood, trade and intelligence sharing.

"May's rushed diplomacy with Trump has been foolish: what has she actually got out of the relationship so far?" said one senior European diplomat in London, who spoke on condition of anonymity.