America has built up a glut of cheese so big that every person in the country would need to eat an extra 3 pounds this year to work it off.

And it isn’t just cheese. The growing stacks of cheddar, which can be kept frozen for years, and other cheeses such as feta, which can be stored for only a couple of months, are just the tip of a surplus of U.S. agricultural products that is swamping markets for grains, meat and milk.

Supplies...