We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. As an application of our techniques, we extend the notion of realisation numbers from rigidity theory to rotational-symmetric and periodic realisation numbers and we provide combinatorial algorithms to compute them. Finally, we show a number of existing matroid invariants are specialisations of the flip product, including the beta invariant.