We study cluster algebras over F₂. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type A, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose corresponding cluster tori cover the cluster manifold over any field F, but there are also sections giving seeds whose cluster tori do not cover the cluster manifold over any field F F₂.
Melesio et al. (Thu,) studied this question.