Left and right are divided about many things, but there is widespread consensus that the U.S. Senate — which delights in its reputation as “the world’s greatest deliberative body” — is broken. Because of polarization by party and the filibuster, which means 60 votes are effectively required for anything to pass, the Senate is often gridlocked.

There is no shortage of solutions on offer: Calls to out-and-out abolish the filibuster appear with numbing regularity. Others ideas are more ambitious: Why not make representation in the Senate proportional to state population, for instance, giving California 12 senators and Vermont one? Whatever their merits, these proposals face serious opposition and therefore stand little chance of succeeding. Sen. Joe Manchin III (D-W.Va.), a key moderate in a body split 50-50 between the parties, has notably said he won’t even vote to end the filibuster.

Still, there is one incremental reform that would make the Senate work more smoothly — even at this divided moment — and that might appeal even to institutional conservatives such as Manchin. There could be a narrow exception to the filibuster for bills that pass out of committee on a bipartisan basis — that is to say, that were supported by a majority of Republicans and a majority of Democrats on the committee. (Roughly equal numbers of each party typically serve on each committee.) Under the reform we envision, that bill would get fast-tracked to the Senate floor for an up-or-down vote.

Such a provision would preserve the filibuster for contentious legislation — so Republicans need not worry they’d get regularly steamrolled for the next two years — while easing the gridlock that has become the Senate’s defining trait.

The Senate is supposed to operate under a process known as “regular order”: Under the leadership of a chair, committees hold hearings, draft a bill, and hold a markup (where the committee members debate and revise the bill). After the committee reaches a compromise, the bill heads to the floor, where it’s debated by the full Senate.

But regular order isn’t so regular anymore. The Senate majority leader can — and often does — decide not to bring bills that come out of committee to a vote. When he does approve a vote, the 60-vote standard for cutting off debate — once rarely invoked — has become the norm. What’s more, the Senate these days tends to act only when faced with a looming crisis. Party leaders cut big deals behind closed doors, eschewing the committee process and unveiling vast bills just hours before a shutdown or an expiration. Senators are then told to take it or leave it, often without knowing what’s really in the legislation.

The recent stimulus bill is a case in point. Leadership released a 5,593 page bill just hours before the Senate voted on it. Senators could not have understood — much less fully considered — the bill they were voting on.

Senators — or former senators — often say they long for a return to “regular order.” But how do you actually encourage a Senate riven by partisanship to embrace a policy process rooted in cooperation at the committee level and beyond? The reform we describe would productively change the incentives. After all, a senator must invest substantial time to have a chance of influencing an important bill. But if bills that pass committee don’t get to the floor, a senator won’t invest that time. He or she would be better off spending the morning on a cable news show, which at least provides a clip for folks back home.

Under a fast-track procedure, the Senate would have to debate any bill with bipartisan committee support within a set time frame and put it to an up-or-down vote shortly thereafter. No scheduling games. No amendments (including poison pills that scuttle the legislation). No 60-vote threshold. A simple majority vote on the bill as written and voted on by the committee.

Committee chairs would be among the obvious beneficiaries of this proposal. Like all politicians, they want to leave a legacy, and legislating is one path to that goal. In a world where they actually might be able to pass significant laws, chairs will allocate more time to legislating and less on TV (and less time planning “blockbuster” hearings on hot-button topics that also generate lots of media clips).

This rule will also change the incentives for rank-and-file members. Passing bills, and then boasting about it, is an effective way to persuade voters to pull the lever at the next election. But senators will only pursue that route if it is actually feasible — and right now, getting laws passed too often isn’t.

Finally, Congress would probably pass better legislation under a fast-track regime. There’s a policy process, which begins with intensive study of relevant issues in a committee, for a reason: It forces senators and their staff to pay attention to the nuts-and-bolts of the legislation. As former senator Bob Corker (R-Tenn.) once put it, “a committee markup actually causes members to understand what’s in a bill.”

It’s also a mathematical certainty that some worthy proposals can garner a majority but won’t reach the 60-vote filibuster standard — and, historically, the filibuster hasn’t been used on every bill. This reform would crack open the door for useful bills that fall just short of that level of support (but do have some backing from both parties).

The reform would also reduce the incentives to play pure partisan politics with bills. Often, a shrewd leader such as Sen. Mitch McConnell (R-Ky.) will engage in “strategic partisanship” — opposing a bill simply because the other party supports it. But committee chairs, and members of committees, may have a different view: They may actually believe in the policy change they’ve devoted considerable time to. And again, they may see personal electoral advantage in getting a bill passed, rather than in defeating the other team through obstruction.

It’s also easier to engage in strategic partisanship when you only need to oppose a few pieces of legislation. When frequent bills become the norm, any single law will seem less salient, less worth fighting over, especially if each one by definition has at least some support in each party.

Fast-tracking bills that pass out of committee with bipartisan support will hardly solve all of the Senate’s problems. It won’t bridge the partisan divide on the most polarizing issues of the day. But it is an incremental solution that will encourage senators to work harder to find consensus and solve discrete problems. And because it preserves the filibuster for major legislation, it might be the kind of reform that the Senate can actually pass, even today.