The Terwilliger algebras of asymmetric association schemes of rank $3$, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension $4k+9$ for some positive integer $k$. It is determined that asymmetric rank $3$ association schemes of order up to $23$ are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this no longer true at order $27$. To distinguish order $27$ asymmetric rank $3$ association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice.