An integrated monopoly, where all 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 consider the possibility of competition in the market for each complement. We present a model with two perfect complements and introduce n imperfect substitutes for one and then for both complements. We prove that, if one complementary good is produced by a monopolist, and if competition for the other complement does not vary the average quality in the market, then an integrated monopoly is still superior. In such case, favoring competition in some sectors, leaving monopolies in others would be detrimental for consumers and producers alike. Competition may be preferred if and only if the substitutes of the complementary good differ in their quality, so that as their number increases, average quality and/or quality variance increases. Results change when competition is introduced in each sector. In this case, if goods are close substitutes, we find that competition may be welfare superior for a relatively small number of competing firms in each sector, even with no quality differentiation.