Combinatorial patchworking: Back from tropical geometry

We show that, once translated to the dual setting of convex triangulations of lattice polytopes, results and methods from Arnal, Renaudineau, and Shaw Ann. H. Lebesgue 4 (2021), pp. 1347–1387, Renaudineau and Shaw Ann. Sci. Éc. Norm. Supér. (4) 56 (2023), pp. 945–980, Jell, Rau, and Shaw Épijournal Géom. Algébrique 2 (2018), p. 27 and Reanaudineau, Rau, and Shaw Real phase structures on tropical manifolds and patchworks in higher codimension, in preparation extend to non-convex triangulations. So, while the translation of Viro’s patchworking method to the setting of tropical hypersurfaces has inspired several tremendous developments over the last two decades, we return to the original polytope setting in order to generalize and simplify some results regarding the topology of T T -submanifolds of real toric varieties.