Abstract. Let G be a digraph. A subset S of V(G) is called an efficient total dominating set if the set of open out-neighbourhoods of vertices of S forms a partition of V(G). We say that G is efficiently open-dominated if both G and its reverse digraph G^- have an efficient total dominating set. We prove that in an efficiently open-dominated digraph G all efficient total dominating sets of G and G^- have the same cardinality; and we investigate the subdigraph of G induced by vertices of an efficient total dominating set in an efficiently open-dominated digraph. We apply our results to tournaments and directed tori.