Sándor Bozóki (2014) has a technique for producing a set of dice that corresponds to any tournament of size $p$ for primes $p\equiv 3\bmod 4$.
But his technique uses many faces: $p(p-1)/2$ for each of the $p$ dice.
For $p=19$, the number of faces is 171 per die!
Can we do better?
Perhaps the voting profile approach will help.