During the first battle of Fallujah, in the spring of 2004, Sunni insurgents fought alongside some Shiite forces against the Americans; by that fall, the Sunnis waged their resistance alone in Fallujah, and they resented the Shiites' indifference.
But by that time, Shiite frustration with Sunnis for harboring Abu Musab al-Zarqawi, the bloodthirsty head of al-Qaeda in Iraq, led some to feel that the Fallujans were getting what they deserved. The cycle of violence escalated from there. When Sunni refugees from Fallujah settled in west Baghdad's Sunni strongholds such as Ghazaliya, al-Amriya and Khadhra, the first Shiiite families began to get threats to leave. In Amriya, Shiites who ignored the threats had their homes attacked or their men murdered by Sunni militias.
This is when sectarian cleansing truly began. Sunni refugees in Amriya seized homes vacated by Shiites. These operations were conducted by insurgents as well as relatives of the refugees. Soon such cleansing had become widespread and commonplace, both out of vengeance and out of its own cruel logic; both sides took part. There was no space left in Iraq for nonsectarian voices. Sunnis and Shiites alike were pushed into the arms of their respective militias, often joining out of self-defense. Shiites obtained lists of the Baath party cadres that were the foundation of Hussein's regime and began systematically assassinating Sunnis who had belonged. Sunni militias that had fought the American occupier became Sunni militias protecting Sunni territory from Shiite incursions and retaliating in Shiite areas. The insurgency became secondary as resistance moved to self-defense. In the Shiite-dominated south, meanwhile, Shiite militias battled each other and the British forces.
http://www.washingtonpost.com/wp-dyn/content/article/2006/05/26/AR2006052601578_pf.html
"When Sunni refugees from Fallujah settled in west Baghdad's Sunni strongholds such as Ghazaliya, al-Amriya and Khadhra, the first Shiiite families began to get threats to leave."
Read between the lines when they talk about "fighters". It was like I said Zarqawi struck the Shia before Fallujah, that mostly stopped when he had a base in Fallujah when it was gone the suicide attacks on Shia started again and al-Qaeda fighters after fleeing Fallujah in early 2005 went to Baghdad and started killing Shia. But, the Badr didn't get into it and this didn't become widespread until late 2005.
However, the Madhi Army didn't get into it until early 2006.