The Ardila-Klivans fan of a matroid provides a direct link between matroids and toric/tropical geometry and has had impressive applications in recent years. In this talk, I will provide a cryptomorphic description of a matroid orientation, known as a real phase structure. Via a generalisation of Viro’s patchworking procedure this provides a link between oriented matroids and real toric varieties and even a homological obstruction to matroid orientation.

Real phase structures are also obtained when tropicalising a real algebraic variety. When the tropicalisation is locally matroidal, I will explain how the tropicalisation together with the real phase structure can be used to recover the topology of a real algebraic variety near the tropical limit. Finally, the topology of these real algebraic varieties, and more general patchworks, can be studied by adapting a spectral sequence from the case of hypersurfaces. This spectral sequence arises from filtrations of the tope space of an oriented matroid.

This talk is based in part on joint work with Allermann & Rau and Yuen.