An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. You can study local systems of vector spaces over this local system of fields. On a 3-manifold, they are are rigid, and the rank one local systems are counted by the Alexander polynomial. On a surface, they come in positive-dimensional moduli (perfect of characteristic p), but they are more "stable" than ordinary local systems in the GIT sense. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On S^1 x R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it.