WASHINGTON — In the heady days after winning control of both congressional chambers, Republican leaders vowed to keep the government funded and to block President Barack Obama from overhauling immigration policies on his own.

But House Speaker John Boehner and Senate Majority Leader Mitch McConnell never crafted a legislative strategy to accomplish those goals. And as events turned against them this week, McConnell said he didn’t know how Boehner would handle a political dilemma heading his way. Boehner, in turn, acknowledged that he and McConnell haven’t talked in two weeks, although their staffs often confer.

For all their victories in the midterm elections, Republicans have been loath to accept the limits of their success. Even with 54 of the Senate’s 100 seats, they still lack enough votes to overcome filibusters, the bill-killing tactic that Democrats are now using against Republicans after years of being on the receiving end.

Senate Republicans confronted that cold reality Tuesday. McConnell publicly conceded he can’t force Senate Democrats to allow action on a contentious House-passed bill. The bill would fund the Department of Homeland Security while also blocking Obama’s 2014 executive order to protect millions of immigrants from deportation even though they are here illegally.

Senate Democrats insist the two issues be separated. McConnell reluctantly yielded Tuesday, and prepared to send House Republicans an unpleasant choice.

They can keep Homeland Security funded beyond this week, inviting fury from anti-Obama groups who want to use the funding as leverage on deportation policies.