Monday's Peshmerga statement accused a group within the PUK of "treason" for assisting Baghdad's advance. The Peshmerga said Baghdad would be made to pay "a heavy price" for triggering "war on the Kurdistan people". He feared the referendum, guaranteed to alienate all the Kurds'allies, would turn out to a political error with similar calamitous consequences.
Tension has steadily mounted between Baghdad and the Erbil-based KRG since September 25, when Iraqis in KRG-controlled areas - and in several disputed territories, including Kirkuk - voted on whether or not to declare political independence.
"They chose their personal interests over Iraq's interests", he added.
A local official in charge of the displaced said tens of thousands, mostly Kurds, were heading out of the city, although at the same time crowds on the streets of Kirkuk's southern outskirts welcomed Iraqi forces.
"This attack, waged by the Iraqi government, Hashd al-Shaabi and forces associated with the Headquarters of the Islamic Revolutionary Guards' Quds force, is in retaliation against the people of Kurdistan who have asked for freedom", the Peshmerga statement said, as reported by the Kurdish news outlet Rudaw.
The elite, USA -trained "Counter Terrorism Service, the 9th armored division and Federal Police have taken control of vast areas of Kirkuk without confrontations", it said, adding that oilfields and Kurdish military positions were captured.
Kurdish Peshmerga fighters seized parts of Kirkuk in 2014 during its battle against Islamic State. Further complicating the matter is that USA special forces and advisers serve with both the Iraqi army and the Peshmerga, setting up a situation whereby American troops could find themselves embedded on opposing sides of the battlefield.
A statement by the US-led worldwide military task force in Iraq described the clashes outside Kirkuk as a "misunderstanding".
"Despite the Kurdistan Regional Government's unfortunate decision to pursue a unilateral referendum, dialogue remains the best option to diffuse ongoing tensions and long-standing issues", he noted.
The military action in Kirkuk helped spur a jump in world oil prices on Monday.
The referendum had faced strong opposition from most regional and global actors (including the U.S., Turkey and Iran), who warned that the poll would distract from Iraq's fight against terrorism and further destabilize the already-volatile region.
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The US has been closely allied to the Kurds in Iraq and Syria, but strongly opposed the independence referendum which it saw as provocative and divisive. Army, police and forces from Iraq's popular mobilization units, which include Shiite militias backed by Iran, have massed in the area, as Kurdish forces furiously dug defenses.
The city of Kirkuk itself remained under Kurdish control, 12 hours after the start of the Iraqi operation, but two routes in and out were under control of the Iraqi forces.
Others grabbed weapons and descended into the streets to help defend Kirkuk against what the Kurdish Region Security Council described as "a major, multi-pronged operation" meant to enter the city and take over a key airbase and oil fields.
Another military commander said: "Kurdish leaders we consider our brothers have agreed to hand over control of North Oil and North Gas company facilities that belong to the state".
Iraqi President Fuad Masum, himself a Kurd, was meeting Sunday with Iraqi Kurd leader Massud Barzani in Dukan in Sulaimaniyah province, officials said.
The withdrawal of part of the Kurdish forces is ultimately a reflection of deep divisions between the Kurdish leaders and their parties, whose rivalry has always been intense.
The PUK had supported a UN-backed plan for negotiations with Baghdad in exchange for dropping the referendum.
Al-Abadi's Cabinet said Sunday that fighters from Turkey's Kurdish insurgency, the PKK, were beginning to appear in Kirkuk, and declared that would be tantamount to an act of war. For years, Turkey has notably described a unilateral move by the Kurds to control Kirkuk as a red line.
During the years of US occupation that ensued, Washington leaned on its Kurdish allies to keep their ambitions in check to avoid triggering another war amid an insurgency by Sunni Muslim Arabs. But the vote crossed a red line in the region, where countries say a unilateral redrawing of the borders can never be permitted. And while Iraq's oil revenues are supposed to be shared, disputes among the provinces have often held up transfers, leading parties to find leverage in holding the fields.
The operation comes as Kurdish Peshmerga fighters gathered to the south of Kirkuk amid an escalating row between the central government in Baghdad and the autonomous Kurdish region of Iraq.