Abstract. There is a product construction which creates face 2-colourable triangulation of K_mn,mn,mn from face 2-colourable triangulations of K_m,m,m and of K_n,n,n. This construction was described in an earlier paper in a topological form. In this paper we describe the very same construction from the point of view of biembedded Latin squares. As an application, we obtain biembeddings in which one square is the Cayley table of Abelian 2-group C^k₂, k>2, and the "other square" has a transversal. This increases the best known lower bound for the number of face 2-colourable triangulations of K_n for an infinite class of values n.