The United States last week launched airstrikes against an Iranian-allied paramilitary group following a deadly attack on an Iraqi base housing US troops -- but rocket fire has continued unabated.

In a phone call with Iraq's caretaker prime minister, Adel Abdel Mahdi, Pompeo said that Baghdad "must defend coalition personnel" who are officially deployed as part of the campaign against the Islamic State group.

"Secretary Pompeo noted that America will not tolerate attacks and threats to American lives and will take additional action as necessary in self-defense," the State Department said in a statement.

A fresh spate of rockets -- brazenly fired in the daytime -- targeted the crowded Taji air base north of Baghdad on Saturday, wounding three members of the US-led coalition.

None of the attacks have ever been claimed but the US has blamed hardline Shiite paramilitary groups, which are allied with neighboring Iran and incorporated into the Iraqi state.

On Wednesday, a similar rocket attack at Taji killed two American personnel and a British soldier in the deadliest such incident at an Iraqi base in years.

The US responded Friday with airstrikes on arms depots it said were used by the faction Kataeb Hezbollah and destroyed part of the under-construction airport in the Shiite shrine city of Karbala.

Iraq's military said that five members of its security forces and one civilian were killed, none members of the targeted paramilitary group.

Iraq -- which has long feared getting caught in spiraling US-Iran tensions -- denounced the "American aggression" and said it would lodge a complaint with the UN Security Council.

The conflict dramatically escalated in early January when the United States killed Iran's best-known general, Qassem Soleimani, in a drone strike at the Baghdad airport.

Baghdad responded by urging US forces to leave but Washington has refused, with Pompeo saying that Iraqi leaders privately wanted troops to stay and President Donald Trump threatening sanctions if US forces are booted out.