President Barack Obama used Washington’s grandest stage—the State of the Union speech—to announce negotiations with Europe aimed at creating the world’s largest free trade agreement. Just weeks later, there are signs that old agriculture disputes could be deal-killers. European Union leaders don’t want the negotiations to include discussions on their restrictions on genetically modified crops and other regulations that keep U.S. farm products out of Europe. But Obama says it’s hard to imagine an agreement that doesn’t address those issues. Powerful U.S. agricultural lobbies will do their best to make sure Congress rejects any pact that fails to address the...