Several algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, to solve the linear systems that arise in their implementations.  An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems, for which there is no need to leave the real axis.  Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective.