Finding forest-orderings of tournaments is NP-complete

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