Abstract. We further investigate a product construction for triangulations by complete tripartite graphs. That construction takes two face 2-colourable triangulations by K_m,m,m and K_n,n,n to obtain a face 2-colourable triangulation by K_mn,mn,mn. The original version works when the triangulation of K_n,n,n has a parallel class of triangles in one colour and our generalization is in using a parallel class that may contain triangles from both colour classes.

Since face 2-colourable triangulations by complete tripartite graphs correspond to biembeddings of Latin squares, we describe the final embedding of K_mn,mn,mn from the viewpoint of underlying Latin squares. Next, we prove a few lemmas about the generalized/original product construction. Using these lemmas, we prove that one specific embedding obtained by the generalized construction, and present in the paper, is nonisomorphic with an embedding obtained by any of the previous versions (by our knowledge there are just two) of product construction. That is, we prove that the generalized construction describes a strictly larger class of embeddings.