  Suppose one needs to change the direction of at least $\epsilon n^2$ edges of
an $n$-vertex tournament $T$, in order to make it $H$-free. A standard
application of the regularity method shows that in this case $T$ contains at
least $f^*_H(\epsilon)n^h$ copies of $H$, where $f^*_H$ is some tower-type
function. It has long been observed that many graph/digraph problems become
easier when assuming that the host graph is a tournament. It is thus natural to
ask if the removal lemma becomes easier if we assume that the digraph $G$ is a
tournament.
  Our main result here is a precise characterization of the tournaments $H$ for
which $f^*_H(\epsilon)$ is polynomial in $\epsilon$, stating that such a bound
is attainable if and only if $H$'s vertex set can be partitioned into two sets,
each spanning an acyclic directed graph. The proof of this characterization
relies, among other things, on a novel application of a regularity lemma for
matrices due to Alon, Fischer and Newman, and on probabilistic variants of
Ruzsa-Szemer\'edi graphs.
  We finally show that even when restricted to tournaments, deciding if $H$
satisfies the condition of our characterization is an NP-hard problem.
