  The self-dual spaces of polynomials are related to Bethe vectors in the
Gaudin model associated to the Lie algebras of types B and C. In this paper, we
give lower bounds for the numbers of real self-dual spaces in intersections of
Schubert varieties related to osculating flags in the Grassmannian. The higher
Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite
Hermitian form. Our bound comes from the computation of the signature of this
form.
