While Washington focuses on Iran-U.S. nuclear talks, the Islamic Republic is making a major but little-noticed strategic advance. Iran’s forces are quietly occupying more of Iraq in a way that could soon make its neighbor a de facto Shiite satellite of Tehran.

That’s the larger import of the dominant role Iran and its Shiite militia proxies are playing in the military offensive to take back territory from the Islamic State, or ISIS. The first battle is over the Sunni-majority city of Tikrit, and while the Iraqi army is playing...