Maybe we both misunderstood, but I read the OP's "which eigen values ... are real" as a question about some of them, not all of them.
It might be possible to give an answer if the eigenproblem represents a physical system. For example the eigenvalues of a damped multi-degree-of-freedom oscillator, with an arbitrary damping matrix, represent the damped natural freuqencies on the s-plane, therefore they are all complex except for zero-frequency (rigid body motion) modes. Also, the sign of the real part of the eigenvalues shows whether the mode is damped, undamped, or unstable (i.e. it gains energy from outside the system).
But I don't know how to turn that "physics insight" about a particular physical system into a mathematical way to characaterize the matrix.