  In this paper we study a singular Finsler double phase problem with a
nonlinear boundary condition and perturbations that have a type of critical
growth, even on the boundary. Based on variational methods in combination with
truncation techniques we prove the existence of at least one weak solution for
this problem under very general assumptions. Even in the case when the Finsler
manifold reduces to the Euclidean norm, our work is the first one dealing with
a singular double phase problem and nonlinear boundary condition.
