  The number of positive solutions of a system of two polynomials in two
variables defined in the field of real numbers with a total of five distinct
monomials cannot exceed 15. All previously known examples have at most 5
positive solutions. Tropical geometry is a powerful tool to construct
polynomial systems with many positive solutions. The classical combinatorial
patchworking method arises when the tropical hypersurfaces intersect
transversally. In this paper, we prove that a system as above constructed using
this method has at most 6 positive solutions. We also show that this bound is
sharp. Moreover, using non-transversal intersections of tropical curves, we
construct a system as above having 7 positive solutions.
