On Saturday, protesters set fire to the entrance of a shrine in Najaf as security forces fired tear gas to disperse them.
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Video footage showed a doorway to the Hakim shrine blazing as protesters cheered and filmed it on their mobile phones.
Iraqis continued to take to the streets of Baghdad and the country's Shia-majority south on Sunday to demand more jobs, an end to endemic corruption, and improved public services.
Security forces killed one protester and wounded nine others near a bridge in the capital on Sunday, police and a medical source said.
In Nasiriyah, two protesters died on Sunday from wounds sustained in previous clashes with security forces, the Reuters news agency reported.
'Iran out of Iraq'
Angry crowds in Najaf set fire to the Iranian consulate last Wednesday, chanting “Iran out of Iraq.”
Iran is seen by many Iraqis as a supporter of the ruling establishment in Iraq.
Staff at the consulate managed to flee just before protesters stormed the building.
Earlier in November, the Iranian consulate in Karbala was targeted by protesters.
Iraq's foreign ministry said the earlier attacks on Iranian diplomatic buildings were "aimed at damaging the historical relations between Iraq and Iran".
Iran’s foreign ministry has blamed Baghdad for failing to protect the consulate and has demanded decisive action against "aggressors".
The latest unrest, which began on 1 October, poses the biggest challenge for Iraq since the Islamic State (IS) group seized swathes of Iraqi and Syrian territory in 2014.
It pits the mostly young, disaffected and unemployed against a government backed by Iran and accused of squandering Iraq's oil wealth, while infrastructure and living standards deteriorate.