LONDON — On his poultry operation outside Manchester, England, Michael Bailey used to raise turkeys and sell them at Christmastime. The whole bird.

But most British customers now want their meat cut up, boneless and, most vexingly to farmers, white, not dark. So Mr. Bailey breaks down his birds, leaving behind a pile of unwanted thighs and drumsticks.

“We’ve got a high-value turkey, and half of it — well, nearly half of it — they don’t want,” he said. “This causes us a major problem.”

For decades, the answer to that problem was the European Union: a frictionless market with idiosyncratic tastes, in which eastern countries crave the dark meat that Britons do not. Along with the turkey leftovers, dark-meat chicken and aging pigs from British farms often land on plates in Eastern Europe, while white meat travels the opposite way. Britons can eat what they want while farmers export what they do not.