The Shia thugs could very well have a large Iranian component rather than being Iraqi. And the jihadis rely on foreign leaders and foreign suicide bombers.
Ledeen writes that the foot soldiers could be more foreign than I've suspected:
Many Iraqis went to Iran during the Iran-Iraq war, where they were trained/indoctrinated by the mullahs for twenty-plus years. We're talking about several million people, not a few cadres. Some of them, along with children, were sent into Iraq to fight us. It's very misleading to simply call them "Iraqis." Maybe they—and their children even more so—should be called "Iranians of Iraqi origin," or "Iranian agents" or some such.
Groups like AQI, along with shi'ite militias, have a strong "foreign" component well beyond the foreigners who run them.
If true, cutting off enemy support for the sectarian killing could result in a fast payoff in reducing killings. It is true that foreign interference takes advantage of the sectarian divisions, but without the foreign interference the sectarian divisions would not result in nearly as much bloodshed.
Ultimately, the biggest surprise of this war has been the willingness of Syria and Iran to fight us in Iraq and our willingness to let Iran and Syria get away with this strategy.