  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.
