Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$?
In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial polytope that cannot be realized as the convex hull of rational vertices? (Such things do exist; see, e.g., p.94 of Gr\"{u}nbaum's Grünbaum's Convex Polytopes.) The vertex sets of the faces of a convex polytope certainly form the flats of a matroid, but it's not clear to me why the same matroid could not be realized by affine dependences of a set of points not in convex position.
