The notion of \(SL_2\)-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic \(SL_2\)-tilings that contain a rectangular domain of positive integers. Every such \(SL_2\)-tiling corresponds to a pair of frieze patterns and a unimodular \(2\times2\)-matrix with positive integer coefficients. We relate this notion to triangulated \(n\)-gons in the Farey graph.