An integrated monopoly, where two complements forming a composite good are offered by a single firm, is typically welfare superior to a complementary monopoly. This is ‘the tragedy of the anticommons’. We analyse the robustness of such result when competition is introduced for one or both complements. Particularly, competition in only one of the two markets may be welfare superior to an integrated monopoly if and only if the substitutes differ in their quality so that, as their number increases, average quality and/or quality variance increases. Then, absent an adequate level of product differentiation, favouring competition in some sectors while leaving monopolies in others may be detrimental for consumers and producers alike. Instead, competition in both markets may be welfare superior if goods are close substitutes and their number in each market is sufficiently high, no matter the degree of product differentiation.