  We study the problem of counting real simple rational functions $\varphi$
with prescribed ramification data (i.e. a particular class of oriented real
Hurwitz numbers of genus $0$). We introduce a signed count of such functions
that is invariant under change of the branch locus, thus providing a lower
bound for the actual count (which does depend on such change). We prove
(non-)vanishing theorems for these signed counts and study their asymptotic
growth when adding further simple branch points. The approach is based on the
works of Itenberg and Zvonkine (arXiv:1609.05219) which treat the polynomial
case.
