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Ardila and Develin's paper on tropical oriented hyperplane arrangements andtropical oriented matroids defines tropical oriented matroids and conjectures abijection between them and triangulations of products of simplices$\Delta_{n-1} \times \Delta_{d-1}$. Oh and Yoo recently confirmed thisconjecture; however, neither group addressed the case of hyperplanes that arenot in generic position. These non-generic arrangements do not correspond totropical oriented matroids, but they encode information about subdivisions of$\Delta_{n-1} \times \Delta_{d-1}$. This note considers the non-generic caseand presents some preliminary results in the area.

Non-Generic Tropical Hyperplane Arrangements and the Secondary polytope of $\Delta_{n-1} \times \Delta_{d-1}$