“Iranian influence is dominant,” said Hoshyar Zebari, who was ousted last year as finance minister because, he said, Iran distrusted his links to the United States. “It is paramount.”

The country’s dominance over Iraq has heightened sectarian tensions around the region, with Sunni states, and American allies, like Saudi Arabia mobilizing to oppose Iranian expansionism. But Iraq is only part of Iran’s expansion project; it has also used soft and hard power to extend its influence in Lebanon, Syria, Yemen and Afghanistan, and throughout the region.

Iran is a Shiite state, and Iraq, a Shiite majority country, was ruled by an elite Sunni minority before the American invasion. The roots of the schism between Sunnis and Shiites, going back almost 1,400 years, lie in differences over the rightful leaders of Islam after the death of the Prophet Muhammad. But these days, it is about geopolitics as much as religion, with the divide expressed by different states that are adversaries, led by Saudi Arabia on one side and Iran on the other.

Iran’s influence in Iraq is not just ascendant, but diverse, projecting into military, political, economic and cultural affairs.

At some border posts in the south, Iraqi sovereignty is an afterthought. Busloads of young militia recruits cross into Iran without so much as a document check. They receive military training and are then flown to Syria, where they fight under the command of Iranian officers in defense of the Syrian president, Bashar al-Assad.

Passing in the other direction, truck drivers pump Iranian products — food, household goods, illicit drugs — into what has become a vital and captive market.

Iran tips the scales to its favor in every area of commerce. In the city of Najaf, it even picks up the trash, after the provincial council there awarded a municipal contract to a private Iranian company. One member of the council, Zuhair al-Jibouri, resorted to a now-common Iraqi aphorism: “We import apples from Iran so we can give them away to Iranian pilgrims.”