LONDON/WASHINGTON (Reuters) - Trade will dominate the first talks between the new leaders of the United States and Britain this week, with both hoping commitments to a future deal will redefine their ‘special relationship’ in a new world order.

For British Prime Minister Theresa May - who will be the first foreign leader to meet new U.S. President Donald Trump - even a simple promise to deepen trade ties could strengthen her hand in divorce talks with the European Union.

Trump might use the meeting to go some way to winning concessions from Britain and bolster his vision of the United States exporting its way to prosperity.

But for both, the road to any firm trade deal is littered with pitfalls and could end up causing strains on the historically close relations between the countries, ties that have been almost driven as much by the personalities of their leaders as national interests.

Differences over genetically modified food, on meat production and public-sector procurement, and fears in Britain that U.S. companies might want to buy into its prized public health service could all hamper any swift movement on a deal.

Plus, while Trump has said a deal can be done “very quickly”, both he and May both say they will put their respective countries’ interests first.

May will meet Trump in Washington on Friday after stopping off in Philadelphia to meet senior Republican leaders from Congress at a retreat the day before.

“So as we rediscover our confidence together - as you renew your nation just as we renew ours - we have the opportunity, indeed the responsibility, to renew the special relationship for this new age,” May will say in Philadelphia on Thursday.

“We have the opportunity to lead, together, again.”

The prime minister will also underline areas where she says cooperation is vital, in defense and security both bilaterally and through NATO, and on Syria.

But it is trade where she hopes to “establish the basis for a strong and productive working relationship”.

It is not yet clear, however, whether Friday’s meeting with Trump will yield a clear shape for future ties. A British government source, who spoke on condition of anonymity, signaled that May’s team was taking a cautious approach, first wanting to get to know Trump’s negotiators and find out what a “quick” trade deal looked like.

‘FUND OF GOODWILL’

The prime minister will be keen to press her Brexit message that she wants to build a “truly global Britain”. But with the EU clear that Britain must not sign trade deals with other countries until it has left and British officials expressing concern over Trump’s shift towards protectionism, May will probably be reluctant about making any binding commitments.

Trump has played up traditionally close ties with Britain, distancing himself from his predecessor Barack Obama who said the country would be at “the back of the queue” for a trade deal with the United States if it left the EU.

And London has made a strong play to court Trump after an initial diplomatic glitch when, soon after his U.S. election victory, he irritated UK officials by meeting British anti-EU campaigner Nigel Farage, a critic of May, and saying he would be a good choice for Britain’s ambassador to Washington.

Following a secret trip by May’s two most senior aides to the United States in December, British foreign minister Boris Johnson met Trump’s close advisers this month and told parliament he had found a “huge fund of goodwill” for Britain.

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But experts and former officials suggest that goodwill may run out fast, not only on trade, but over other areas where Trump and May have potential to disagree, such as climate change, the NATO military alliance and the Iranian nuclear deal.

“Beware of Donald Trump bearing gifts,” Mark Malloch Brown, a former British government minister and United Nations deputy secretary-general, told Reuters, suggesting that the U.S. president was not a “fan” of trade deals.

Trump formally withdrew the United States from the Trans-Pacific Partnership trade deal this week and is also working to renegotiate the North American Free Trade Agreement to provide more favorable terms.

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BACK-FOOT BRITAIN

Britain has yet to launch its exit negotiations with the European Union, promising to do so before the end of March, and faces some of the toughest talks it has waged since World War Two to end a relationship of more than 40 years.

May says she will leave the EU’s single market, instead focusing on winning a free trade deal with the bloc and agreements with other countries.

By making clear she will cut ties with the EU unless she wins a good deal, some experts say she has handed the United States and other countries the upper hand in any future talks.

“If the EU-UK discussion goes badly, the UK is going to be left in a position of being very exposed, and wanting to find new partners quickly. Who’s going to be sitting there at that point? The U.S.,” said a former trade official in the Obama administration, who spoke on condition of anonymity.

After a U.S.-EU trade deal, the Trans-Atlantic Trade and Investment Partnership (TTIP), ground to a halt last year, Washington might press for Britain to drop its resistance to U.S. genetically modified foods and to smooth over regulatory differences for product safety, food and pharmaceuticals.

The two sides could also find a way to reduce regulation on financial services, although with New York and London as rival centers, any such agreement could be difficult, the former Obama trade official said.

In Britain, opposition lawmakers have already challenged May on whether she will lower health and safety standards to allow imports of U.S. beef that contains growth hormones, chicken washed in chlorinated water and genetically modified organisms.

“We will be looking for a UK-U.S. trade deal that improves trade between our two countries,” May told parliament on Wednesday. “And I can assure ... that in doing that we will put UK interests and UK values first.”