  In the 1990's, Itenberg and Haas studied the relations between combinatorial
data in Viro's patchworking and the topology of the resulting non-singular real
algebraic curves in the projective plane. Using recent results from Renaudineau
and Shaw on real algebraic curves near the non-singular tropical limit, we
continue the study of Itenberg and Haas inside any non-singular projective
toric surface. We give new Haas' like criteria for patchworking non-singular
real algebraic curves with prescribed number of connected components, in terms
of twisted edges on a non-singular tropical curve. We then obtain several
sufficient conditions for patchworking non-singular real algebraic curves with
only ovals in their real part. One of these sufficient conditions does not
depend on the ambient toric surface. In that case, we count the number of even
and odd ovals of those curves in terms of the dual subdivision, and construct
some new counter-examples to Ragsdale's conjecture.
