Transitivity is a central, generative principle in social and other complex networks, capturing the tendency for two nodes with a common neighbor to form a direct connection. We propose a new model for highly dense, complex networks based on transitivity, called the Iterated Local Model Tournament (ILMT). In ILMT, we iteratively apply transitivity to form new tournaments by cloning nodes and their adjacencies, and either preserving or reversing the orientation of existing arcs between clones. The resulting model generates tournaments with small diameters and high connectivity as observed in real-world complex networks. We analyze subtournaments or motifs in the ILMT model and their universality properties. For many parameter choices, the model generates sequences of quasirandom tournaments. We also study the graph-theoretic properties of ILMT tournaments, including their cop number, domination number, and chromatic number. We finish with a set of open problems and variants of the ILMT model for oriented graphs.