The obvious area for Republicans to actually agree to do something is with immigration. Sure their base hates it, but their base hates everybody and everything. Can they bring themselves so support a path to citizenship that isn't so difficult/expensive/punitive as to be meaningless?
Opposition to this stuff is in part racism, but it's also based on the general belief in wingnuttia that the blahs and the brahs, even the illegal brahs, have access to some secret super generous welfare system that white people can't get access to. As Craig T. Nelson said, "I have been on food stamps and welfare no one helped me out."