  In 2008 Bella, Olshevsky and Prasad proved that the flipped orthogonal (FO)
Jordan bases of H-selfadjoint matrices are Lipschitz stable under small
perturbations. In 2022, Dogruer, Minenkova and Olshevsky considered the real
case, and proved that for real H-selfadjoint matrices there exist a more
refined bases called FOCS bases. In addition to flipped orthogonality they also
possess the conjugate symmetric (CS) property. In this paper we prove that
these new FOCS bases are Lipschitz stable under small perturbations as well. We
also establish the Lipschitz stability for the classical real canonical Jordan
bases.
