With the World Trade Organisation and the European Commission due to meet in emergency session tomorrow, the two committed themselves to looking for "compatibility" rather than confrontation.
At the end of the Second World War, Winston Churchill saw Britain's future in the intersection of three circles: the Commonwealth, Europe and the Atlantic community. Who would have thought that this grand vision would be shattered by the banana?
The present trade crisis may seem risible to those unaffected by the sanctions imposed in Washington last week. But it is possible to see in it every dilemma of post-war British foreign policy in miniature.
Britain is intertwined with the Commonwealth through history, with America through strategic choice and with Europe by virtue of geography and political ambition. When these regions are moving in different directions, as they are, Britain finds itself in a painful position, with hundreds of British workers in factories making cashmere sweaters facing unemployment. Britain wants to help the Caribbean banana growers; but can it do that at the expense of other British interests?
Churchill's neat formula was designed to maintain British prestige and power: each allegiance was designed to counter-balance the other.
Every post-war prime minister has discovered, usually to their cost, that it was simply impossible to maintain a balance. Eden chose Empire and France over America in 1956 when he invaded Egypt to seize the Suez Canal, and lost his job. Since then, the order of preference has usually found America at the top and the Commonwealth at the bottom, with Europe placed uncomfortably in the middle.
Tony Blair had looked like the model Prime Minister. He had warm and sunny relations with Washington, had improved Britain's ties to Europe no end, and, unlike Margaret Thatcher, is a firm supporter of the Commonwealth, remodelled as a genuinely independent block outside British control.
Now a bunch of fruits of the genus musa (family Musaceae) look set to overturn everything. The banana regime exists because member states in Europe - especially Britain and France - want to help their former (and in some cases existing) colonies. Both allow their bananas preferential access, so more are imported than would be under truly free trade.
The objection stems from American companies that grow bananas in Central and South America which have enough influence in Washington to get their way. Britain is caught in the middle.
British diplomats in Washington spend a lot of time trying to ensure such conflicts do not occur. But they do, and many involve the remains of Empire.
On nearly every issue Britain and the US are closer than any two countries, but America has little interest in Britain's remaining post-colonial pretensions at the best of times, as it showed over Suez. The US invaded Grenada apparently without a thought to the fact that the island was part of the Commonwealth, with the Queen as head of state. That was one of the few points of friction between Mrs Thatcher and Ronald Reagan. And many in his administration thought America should have backed Argentina in its fight against Britain over the Falklands.
But when transatlantic crises involve Europe, Britain usually aligns tacitly with the US. In nearly every other dispute - especially those concerning agriculture - we are seen as the enemy within, sympathising with Washington and its desire to crack EU protectionism. Britain shares a preference for trade, over continental Europe's more mercantilist tendencies. But things may be changing.
To see bananas as the motor force of history, dividing London and Washington, seems ridiculous, and it is. But there are at least two other disputes where divisions are clear: a complex row over aircraft noise that could see Concorde banned; and Iran, where British, French and Italian oil companies are defying American sanctions. In both cases, British and EU interests are congruent. As the single market becomes a reality and corporate tie- ups proliferate across the Channel, that may be increasingly the case. And American support for free trade may become less easy to guarantee as its trade deficit soars.
But there are other reasons why Britain feels it should back the European line on bananas. The central one is that the problems of the Caribbean producers are largely its fault. The island states would not exist in their present form were it not for the rivalries of the 18th century, which pitted Britain against Spain and France. That is how Britain came to rule over places called Anguilla, Antigua and Dominica.
When the French Wars ended in 1815, Britain effectively ditched the oldest part of the Empire. It dealt a body blow to the region's economy when it chose free trade in the 1840s, getting rid of the measures which had protected Caribbean exports to the homeland. It wanted to encourage trade with the new United States of America, not the moribund monocultures of the West Indies.
Ironically, this turn towards free trade helped to boost British textiles, turning the north of England into the workshop of the world. Little remains of that now, apart from the cashmere sweater producers of the Borders. Caribbean economies, in near-terminal decline, had to turn to alternative products, such as bananas.
Britain then largely ignored the Caribbean until the late 1940s, when it belatedly realised there was growing political dissent and dire poverty. It started to develop their economies - such as boosting banana production, and encouraging exports to Britain.
The argument is rich with historical ironies; unfortunately, neither the Caribbean nor the Borders are rich in anything much, apart from cashmere sweaters and bananas.
Now the banana dispute is real and something is going to have to give way. This time, the weak link may be the apparently unbreakable one between London and Washington.