"Unlike the U.S., which thinks its interests are secured by causing rift and division in the region, the Islamic Republic of Iran believes its interests are met through the cooperation of countries in the Middle East", he went on to say.
The two had earlier sparred over Iranian influence in Iraq.
At his meeting with Abadi in Baghdad, Tillerson urged the Iraqi government and Masoud Barzani's Kurdistan Regional Government in Erbil to resolve their conflict on Kurdish self-determination and disputed territories through dialogue.
During the meeting, also attended by Prime Minister Sherif Ismail, Abadi offered his country's condolences to Cairo over the terrorist attack that left several Egyptian policemen dead and wounded. "Shameful U.S. foreign policy dictated by petrodollars", railed Iranian Foreign Minister Mohammad Javad Zarif on Twitter, disingenuously portraying the PMF as a grass-roots home-grown counter-insurgency force.
The forces of the PMU, also known as Hashd al-Shaabi, are Iraqi citizens and have sacrificed themselves to defend their country, the statement read.
Tillerson arrived on Monday hours after the Iraqi government rejected his call to send home the Popular Mobilisation, an Iran-backed force that helped defeat Islamic State and capture the Kurdish-held city of Kirkuk.
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President Abdel Fattah El Sisi on Sunday22/10/2017 received Iraqi Prime Minister Haider al Abadi. Saudi Arabia and Iraq have been estranged ever since the latter invaded Kuwait in 1990 under the leadership of former dictator Saddam Hussein.
"We are concerned and a bit sad", Tillerson said in his opening remarks.
"We don't want to enter into any battle against any Iraqi component", Abadi said. "No side has the right to intervene in Iraq's affairs or decide what Iraqis should do".
The Associated Press describes a us strategy to create a "new axis" of power in the Middle East that "unites Saudi Arabia and Iraq as a bulwark against Iran's growing influence from the Persian Gulf to the Mediterranean Sea".
The United States has over 5,000 troops deployed in Iraq and provided critical air and ground support in the offensive on Islamic State.
The Iraqi government was surprised by Tillerson's suggestion, according to the release.
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