IN DEVOTING 250 of the 6,800 words of his State of the Union address to the fight against “violent extremism,” President Obama offered a boilerplate description of his policy. “Instead of sending large ground forces overseas,” he said, “we’re partnering with nations from South Asia to North Africa to deny safe haven to terrorists who threaten America.” As he spoke, his strategy was crumbling in a nation he failed to mention: Yemen, home to the branch of al-Qaeda that claimed credit for the recent attacks in France and has repeatedly attempted to strike the U.S. homeland.

Since 2012, the United States has tried to build a “partnership” against al-Qaeda with President Abed Rabbo Mansour Hadi, training selected Yemeni counterterrorism units and providing close to $1 billion in aid. Mr. Hadi, in turn, has allowed the CIA and Pentagon to use drone strikes to directly target militants deemed to be plotting against the United States. Though cited as a model by Mr. Obama last September, the program has been foundering: Al-Qaeda has mounted an insurgency that gained territory. Meanwhile, Mr. Hadi’s government has come under assault from rebels known as the Houthis, who are members of a Shiite sect.

Backed by Iran, the Houthis took control of most of the capital, Sanaa, several months ago. This week, dissatisfied with Mr. Hadi’s failure to deliver on their political demands, they besieged the presidential palace, abducted Mr. Hadi’s chief of staff and made the president a prisoner in his home. On Wednesday Mr. Hadi was reported to have capitulated to the rebels’ demand that a proposed constitutional reform be rewritten to give the Houthis, who represent less than a third of Yemen’s population, more power. On Thursday, he was reported to have resigned.

Yemen’s politics are beyond byzantine, but it’s not hard to understand what is happening here: Iran’s clients, who happen to have adopted the same “death to America” slogan as Hezbollah, are imitating the Lebanese Shiite party’s strategy of establishing a chokehold on a weak Sunni government. It’s worth noting that Iran has not felt constrained from sponsoring this proxy aggression even as it continues to negotiate about its nuclear program with the United States and its allies. The Obama administration, meanwhile, claims that it dare not support Syrian rebels against the Iranian-backed regime in Damascus, or allow Congress to approve future sanctions, lest Tehran be spooked into abandoning the talks.

The administration may be hoping that the Houthis will help defeat al-Qaeda, a Sunni organization the Houthis regard as an enemy. Predictably, however, Iran’s clients have denounced the U.S. presence in the country. They could force the shutdown of training and drone operations — or they may trigger a civil war that will make those operations impossible to maintain.

The Yemen mess reveals the weaknesses of Mr. Obama’s “partners” strategy, which has been too narrowly focused on drone strikes and training of specialized units, and not enough on providing security for the population, institution-building and support for moderate political forces. Unfortunately, the president’s cursory and formulaic description of his counterterrorism policies this week, following a year in which jihadist forces and terrorist attacks expanded across the world, suggested that he remains uninterested in correcting his mistakes.