Full waveform seismic inversion (FWI) in the viscoelastic regime entails the task of identifying parameters in the viscoelastic wave equation from partial waveform measurements. Traditionally, one frames this nonlinear problem as an operator equation for the parameter-to-state map. Alternatively, in an all-at-once approach one augments the nonlinear operator by the viscoelastic wave equation as an additional component and considers the states as additional variables. Hence, parameters and states are sought-for simultaneously. In this article, we give a mathematically rigorous all-at-once version of FWI in a functional analytical formulation. Further, the corresponding nonlinear map is shown to be Fréchet differentiable and the adjoint operator of the Fréchet derivative is given in an explicit way suitable for implementation in a Newton-type/gradient-based regularization scheme.