On vector spaces with distinguished subspaces and redundant base

Let V,W be finite dimensional vector spaces over a field , each with n distinguished subspaces, with a dimension-preserving correspondence between intersections. When does this guarantee the existence of an isomorphism between V and W matching corresponding subspaces? The setting where it happens requires that the distinguished subspaces be generated by subsets of a given redundant base of the space; this gives rise to a (0,1)-incidence table called tent, an object which occurs in the study of Butler B(1)-groups.