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The number of positive solutions of a system of two polynomials in twovariables defined in the field of real numbers with a total of five distinctmonomials cannot exceed 15. All previously known examples have at most 5positive solutions. Tropical geometry is a powerful tool to constructpolynomial systems with many positive solutions. The classical combinatorialpatchworking method arises when the tropical hypersurfaces intersecttransversally. In this paper, we prove that a system as above constructed usingthis method has at most 6 positive solutions. We also show that this bound issharp. Moreover, using non-transversal intersections of tropical curves, weconstruct a system as above having 7 positive solutions.

Constructing polynomial systems with many positive solutions using tropical geometry