Consensus appeared likely until late Friday, when some Shiite members raised objections to the draft document and sent it back for more debate.
Resistance was particularly strong among some Shiite members, who represent the country's largest group and have been pushing for quick elections.
With that, according to council members, five male Shiite members of the council stormed out of the room.
The two-thirds vote requirement effectively gives the Shiite members final say over any decision, as long as they remain united.
Other Shiite members repeated similar stories, and the number of Shiite hostages steadily grew, from 60 to 80 to as high as 200.
Other Shiite members of the assembly defended their action.
Many of the Shiite members now support immediate direct elections.
Protests by women and relatively secular blocs on the constitutional committee, like the Kurds, may force Shiite members to tone down the religious language.
"This is a disgrace," said Raja Kuzai, a Shiite member of the council.
"We worry about the situation," said Abbas Noyan, a Shiite member of Parliament.