IN THE DAYS since President Obama’s victory, it has become an instant truism that Republicans, desperate to halt their free fall with Hispanic voters, need to make a sweeping deal on immigration reform. But no matter how strong the political impetus, fixing the nation’s broken immigration system, which proved too tough for Congress in the past decade, remains a subject for hard bargaining. And no part of the deal will be harder than resolving the status of illegal immigrants.

It’s obvious — though Mitt Romney had to learn it the hard way — that 11 million undocumented residents will not “self-deport,” nor should they. Two-thirds of them are in the work force; many of those have been in this country for a decade or more; and some have children, born here, who are American citizens. They are a pillar of the American economy. The fair and sensible solution is to grant them a path to citizenship.

Still, some fudging at the margins may be necessary to reach an agreement, especially given the depth of grass-roots Republican hostility to “amnesty.” A creative compromise could take many forms, such as conferring legal status on undocumented immigrants and removing the threat of deportation for those with no criminal history but postponing the question of citizenship for a finite number of years. That wouldn’t be ideal — it could be portrayed as a program of second-class citizenship. But it would put an end to state legislation designed to harass undocumented residents and would allow immigrants to lead open, secure lives.

The other components of an immigration deal will be only marginally easier, but without a comprehensive bargain that includes the “smaller” pieces the two parties are unlikely to resolve the central question of the 11 million. Democrats should be prepared to agree to additional enforcement measures, especially in establishing a watertight system for employers to verify that job applicants are in the country legally.

Legal immigration will also have to be changed. Among the urgent priorities is attracting skilled workers and especially students who receive advanced degrees at American universities in science, technology, engineering and math. Too often, they are turned away; that is lunacy.

At the same time, businesses must have timely access to adequate numbers of seasonal and agricultural workers, and U.S. citizens’ relatives who wish to immigrate should not languish for years. Both parties will have to compromise on the mechanisms by which annual quotas are set.

In the past, much of the jockeying around solving the immigration problem has been about who would get blame or credit. Even now, some Republicans are openly fearful that a deal would simply cement Democrats’ electoral advantage among Latinos — and, possibly, create millions of new Hispanic voters. Some Democrats might rather milk the status quo, which has helped them until now, than make tough compromises.

Both parties would be wise to realize that standing pat carries its own risks. Every poll suggests that large majorities of Americans want the immigration system fixed. Congress should heed that message.