  Given a class of (undirected) graphs $\mathcal{C}$, we say that a FAS $F$ is
a $\mathcal{C}$-FAS if the graph induced by the edges of $F$ (forgetting their
orientations) belongs to $\mathcal{C}$. We show that deciding if a tournament
has a $\mathcal{C}$-FAS is NP-complete when $\mathcal{C}$ is the class of all
forests. We are motivated by connections between $\mathcal{C}$-FAS and
structural parameters of tournaments, such as the dichromatic number, the
clique number of tournaments, and the strong Erd\H{o}s-Hajnal property.
