Reid: There will be tax hikes

Reid added that without an agreement on both tax hikes and spending cuts, the committee would deadlock and trigger blunt across-the-board-cuts to defense spending and social spending. This was a key provision in the agreement that both sides said would help create an incentive for both parties to work together and compromise on further cuts.

"The trigger is going to kick in," said Reid. "The only way we can arrive at a fair arrangement for the American people with this joint committee is to have equal sharing."

Reid spoke in response to comments from Senate Minority Leader Mitch McConnell (R-Ky.), who implied than Republicans will seek to stack the committee with appointees committed to focus on spending reductions, not tax hikes.

"So much work remains," McConnell said. "Our first step will be to make sure those Republicans who sit on the powerful cost-cutting committee are serious people who put the best interests of the American people and the principles that we've fought for."

Congressional passage of the debt ceiling compromise puts immediate pressure on House and Senate leaders to choose members of the new joint committee.

Under the bill, Reid, McConnell, House Speaker John Boehner (R-Ohio) and Minority Leader Nancy Pelosi (D-Calif.) each get to pick three members of the 12-member committee, and must appoint them no later than 14 days after the bill becomes law, or August 16.

But House and Senate leadership offices indicated Tuesday they had no information yet on how members might be chosen, and when they might be announced.

Once nominated, the joint committee is required to meet no later than 45 days after the bill becomes law, or mid-September. And then, by October 14, the joint committee must make recommendations on how to trim the deficit. The committee is required to vote on a report and proposed legislation by November 23.

-- This story was updated at 2:15 p.m. to add details about the joint committee.