Tropical matroid Schubert varieties and the graded Möbius algebra

We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid $M$ is isomorphic to the graded Möbius algebra ${\mathrm{B}}^{\bullet }(M)$. This yields a geometric model for ${\mathrm{B}}^{\bullet }(M)$, extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.