Genomic conflict is perplexing because it causes the fitness of a species to decline rather than improve. Many diverse forms of genomic conflict have been identified, but this extant tally may be incomplete. Here, we show that the unusual characteristics of the sex chromosomes can, in principle, lead to a previously unappreciated form of sexual genomic conflict. The phenomenon occurs because there is selection in the heterogametic sex for sex-linked mutations that harm the sex of offspring that does not carry them, whenever there is competition among siblings. This harmful phenotype can be expressed as an antagonistic green-beard effect that is mediated by epigenetic parental effects, parental investment, and/or interactions among siblings. We call this form of genomic conflict sexually antagonistic "zygotic drive", because it is functionally equivalent to meiotic drive, except that it operates during the zygotic and postzygotic stages of the life cycle rather than the meiotic and gametic stages. A combination of mathematical modeling and a survey of empirical studies is used to show that sexually antagonistic zygotic drive is feasible, likely to be widespread in nature, and that it can promote a genetic "arms race" between the homo- and heteromorphic sex chromosomes. This new category of genomic conflict has the potential to strongly influence other fundamental evolutionary processes, such as speciation and the degeneration of the Y and W sex chromosomes. It also fosters a new genetic hypothesis for the evolution of enigmatic fitness-reducing traits like the high frequency of spontaneous abortion, sterility, and homosexuality observed in humans.