  In a recent paper we showed that a subspace of a real JBW*-triple is an
M-summand if and only if it is a weak*-closed triple ideal. As a consequence,
M-ideals of real JB*-triples, including real C*-algebras, real JB*-algebras and
real TROs, correspond to norm-closed triple ideals. In the present paper we
extend this result to (possibly non-selfadjoint) real operator algebras and
Jordan operator algebras, where the argument is necessarily different. We also
give simple characterizations of one-sided M-ideals in real operator algebras,
and give some applications to that theory.
