  We study realization spaces of matroids over hyperfields (in the sense of
Baker and Bowler). More precisely, given a matroid M and a hyperfield H we
determine the space of all H-matroids over M. This can be seen as the matroid
stratum of the hyperfield Grassmannians in the sense of Anderson and Davis. We
give different descriptions of these realization spaces (e.g., in terms of
Tutte groups or projective classes), allowing for explicit computations. When
the hyperfield at hand is topological, the realization spaces have a natural
topology. In this case, our models carry the correct homeomorphism type. As
applications of our methods we obtain a theorem on the existence of phased
matroids that are not realizable over the complex field and are not chirotopal,
as well as a result on the diffeomorphism type of complex hyperplane
arrangements whose underlying matroid is uniform.
