For non symmetric systems, a Schur decomposition instead of an eigenvalue decomposition is used internally. This is a preparation for the non-symmetric case.
So Eigen has a Schur decomposition, but the sorting of eigenvalues gets more complicated for schur decomposition, so the rodering of schur vectors would have to be implemented which is really not easy.
For non symmetric systems, a Schur decomposition instead of an eigenvalue decomposition is used internally. This is a preparation for the non-symmetric case.