Lemke and Howson proposed in 1964 a pivot-based algorithm for solving bimatrix games. The year after, Lemke proposed a closely related algorithm for solving a large class of linear complementarity problems. Although Todd extended the Lemke algorithm to oriented matroids in 1984, the extension of the Lemke—Howson algorithm to oriented matroids has not yet been addressed in the literature. Actually, achieving such an extension requires solving a few issues. One of them is to propose a relevant definition of Nash equilibria for oriented matroids. Another one is more challenging, namely finding the counterpart of the following preliminary operation of the classical Lemke—Howson algorithm: the translation of the payoff matrices so as to make them positive. In this work, we show how to overcome these challenges and extend the Lemke—Howson algorithm to oriented matroids. A few applications are discussed, such as a strengthening of a theorem by McLennan and Tourky, a combinatorial interpretation of the orientation of the pivot step in the classical version, and the localization of tropical bimatrix games in the class PPAD. Joint work with Xavier Allamigeon and Stéphane Gaubert.