Coevolution is relentlessly creating and maintaining biodiversity, and therefore has been a central topic in evolutionary biology. Previous theoretical studies have mostly considered coevolution between genetically symmetric traits (i.e., coevolution between two continuous quantitative traits or two discrete Mendelian traits). However, recent empirical evidence indicates that coevolution can occur between genetically asymmetric traits (e.g., between quantitative and Mendelian traits). We examine consequences of antagonistic coevolution mediated by a quantitative predator trait and a Mendelian prey trait, such that predation is more intense with decreased phenotypic distance between their traits (phenotype matching). This antagonistic coevolution produces a complex pattern of bifurcations with bistability (initial state dependence) in a two-dimensional model for trait coevolution. Further, with eco-evolutionary dynamics (so that the trait evolution affects predator-prey population dynamics), we find that coevolution can cause rich dynamics including anti-phase cycles, in-phase cycles, chaotic dynamics, and deterministic predator extinction. Predator extinction is more likely to occur when the prey trait exhibits complete dominance rather than semidominance and when the predator trait evolves very rapidly. Our study illustrates how recognizing the genetic architectures of interacting ecological traits can be essential for understanding the population and evolutionary dynamics of coevolving species.