Fuchsian Riemann-Hilbert problem for “real doubles” of Hurwitz Frobenius manifolds

A solution to the matrix Fuchsian Riemann-Hilbert problem (inverse monodromy problem) corresponding to “real doubles” of Dubrovin's Frobenius structures on Hurwitz spaces is constructed. The solution is given in terms of certain meromorphic differentials integrated over a basis of an appropriate relative homology space of the Riemann surface. The relationship with the solution of Fuchsian Riemann-Hilbert problem for Dubrovin's Hurwitz Frobenius manifolds is established. A solution of the Riemann-Hilbert problem corresponding to deformations of the “real doubles” is also given.