With most votes counted, a bloc headed by cleric Moqtada al-Sadr and another led by a militia leader are ahead, voting officials are quoted as saying.
Saturday's balloting was the first since Iraq declared victory over the Islamic State.
Abadi was widely seen as the front-runner.
PM Abadi heads the Nasr Coalition (Victory of Iraq), its name capitalising on his government's victory over ISIL in 2017. The Shia militia announced it would kill any British troops it saw.
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The Victory Alliance of Prime Minister Haider al-Abadi, who has been backed by the worldwide community, looked to have won in only one province.
Unlike Abadi, a rare ally of both the United States and Iran, Sadr is an enemy of both countries that have wielded influence in Iraq after the US-led invasion that toppled Sunni dictator Saddam Hussein and ushered the Shi'ite majority to power.
Remembered for leading an insurgency against US forces and inciting sectarian bloodshed against the Sunni population, al-Sadr has in recent years sought to recast himself as a populist, railing against corruption and failing services and striking a political alliance with Iraq's secularists and Communist Party. Sadr sought to broaden his regional support, meeting Saudi Crown Prince Mohammed bin Salman in Jeddah a year ago.
The electoral surprise comes with tensions surging between the USA and Iran after Washington's withdrawal from the 2015 nuclear deal, sparking fears of a destabilizing power struggle over Iraq.
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A Sadr-led government could be a big change in Iraq's regional alignment.
The leader of the party winning Muktad al-Sadr is known that in 2004, led a revolt against the American military. But the defeat of Islamic State and Abadi's campaign to eradicate Iraq's rampant corruption improved his standing.
According to partial results, the Marching Towards Reform alliance of Shia cleric Sadr and his communist allies was ahead in six of Iraq's 18 provinces and second in four others. How the vote tally translates into parliamentary seats will be announced later this week, Iraq's election commission said.
The head of the list is Hadi al Ameri, a long-time ally of Tehran, whose forces ended up battling alongside the U.S. to oust the militants. Dozens of alliances ran for office in these elections and months of negotiations are expected before any one alliance can pull together the 165 required seats. Instead, the bloc that wins the most seats will have to bring together a majority by getting the support of smaller alliances.
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