  We prove that, with high probability, any $2$-edge-colouring of a random
tournament on $n$ vertices contains a monochromatic path of length $\Omega(n /
\sqrt{\log n})$. This resolves a conjecture of Ben-Eliezer, Krivelevich and
Sudakov and implies a nearly tight upper bound on the oriented size Ramsey
number of a directed path.
