  We give lower bounds for the numbers of real solutions in problems appearing
in Schubert calculus in the Grassmannian Gr(n,d) related to osculating flags.
It is known that such solutions are related to Bethe vectors in the Gaudin
model associated to gl(n). The Gaudin Hamiltonians are selfadjoint with respect
to a nondegenerate indefinite Hermitian form. Our bound comes from the
computation of the signature of that form.
