  We develop a theory of groups that act acylindrically and non-elementarily on
simplicial trees, which we call acylindrically arboreal groups. We then prove a
complete classification of when graph products of groups and the fundamental
groups of certain hyperbolic $3$-manifolds are acylindrically arboreal, and use
this classification to consider acylindrically hyperbolic groups that have
actions on trees but have no non-elementary acylindrical actions on trees.
