Counting real Galois covers of the projective line

For Galois covers of P-1 of a given ramification type - essentially, a given monodromy group G and branch locus, assumed to be defined over R - we ask: How many covers are defined over R and how many are not? J.-P. Serre showed that the number of all Galois covers with given ramification type can be computed from the character table of G. We adapt Serre's method of calculation to the more refined situation of Galois covers defined over R, for which there is a group-theoretic characterization due to P. Debes and M. Fried. We obtain explicit answers to our problem. As an application, we exhibit new families of covers not defined over their field of moduli, the monodromy group of which can be chosen arbitrarily large. We also give examples of Galois covers defined over the field Q(tr) of totally real algebraic numbers with Q-rational branch locus.