We forgot to post this here in the beginning of the summer… The paper outlines a very general framework for (an) algorithm(s) waiting to be implemented.
Beyond polyhedral homotopies
(Anton Leykin and Josephine Yu)
Abstract: We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.