A visitor in Sana also notices a surprising sense of internal stability. The Houthis are building something akin to a police state — the lack of checkpoints or other markers of security in the capital announcing their effective stranglehold.

Most people in Sana, rightly, consider the United States to be complicit in the war, an enabler of the Saudi-led coalition that wages it. Americans, understandably, would recoil at the Houthis embracing “Death to America” and “Curse the Jews” as their slogans, scrawled as graffiti on the city’s walls. But Sana’s residents warmly welcome the rare American visitor.

The Houthi leadership knows all this — the popular hostility toward the Saudi-led coalition; the remarkable control the movement has achieved — and finds other justifications for self-confidence. Time, they feel, is on their side. Despite formidable military disparities, they have stood up to a coalition of wealthy, powerful states backed and armed by the West.

Front lines are essentially frozen. Their domestic foes, nominally led by the Aden-based, internationally recognized government of President Abd Mansour Hadi, show signs of fracture. The Emiratis recently announced a redeployment that greatly diminishes the risk of a coalition assault on the vital port city of Hudaydah. Anti-Saudi sentiment is rising in the halls of the United States Congress.

Despite their confidence, the Houthis don’t know how or when this war will end. Their singular refrain is that they are ready to talk. Not with Mr. Hadi or his allies, whom they dismiss as “mercenaries” — but with the Saudis, who they claim pull Mr. Hadi’s strings, or with the United States, which they believe pulls the strings of the Saudis.

They offer a road map: They promise to stop cross-border attacks against Saudi Arabia if the Saudis halt attacks against them. They will withdraw from Saudi territory. Riyadh will allow Sana airport to reopen. And they will discuss a longer-term relationship that, they assert, will be closer than the Houthis’ relations with Iran.

They talk in one breath about a peaceful settlement and close ties to Saudi Arabia; in another they warn of surprises that lie in store if Riyadh doesn’t agree to a cease-fire. They take offense at being labeled Iranian proxies, but acknowledge the war brought them closer and offer lackluster denials when asked if Iran supplies them with weapons. At times they emphasize the purely local nature of their fight; at others its more revolutionary, Pan-Islamic identity. Saudi and American officials can be forgiven for being confused.