The importance of studying viscoelastic fibre-reinforced biological materials is underscored by their ubiquity in vital components of the human body, such as muscles, the brain, and reproductive organs, to cite but a few. In particular, the fibres in these tissues are often statistically distributed, and the interplay between alignment and viscoelasticity is important to understand the mechanical properties. To address this issues, we present a comprehensive model for anisotropic viscoelastic biological materials that can handle large deformations. The model, based on the kinematic assumption that the reinforcing fibre structure undergoes affine deformation with the underlying matrix, incorporates fibre dispersion using a generalized structural tensor approach. Furthermore, we consider a strain energy function that integrates elastic and overstress components, corresponding to distinct natural states of the material and hence to a remodelled state which is not necessarily stress-free. Importantly, we recognize that viscous remodelling alters the fibre distribution, thereby inducing a dependence of the overstress energy on a remodelled fibre orientation tensor and not on the reference one. An anisotropic evolution equation for viscous strain is then derived which includes five distinct characteristic times. To test the main features of the model, we firstly show that the evolution of the viscous strain variable can be recast into a referential variational form, facilitating numerical treatment and avoiding any assumption on the viscous spin. The Finite Element implementation is performed by means of a staggered algorithm for the solution of the viscoelastic problem. Our results showcase the model's ability to capture various viscoelastic material behaviours, such as relaxation, recovery, hysteresis, and creep. Hence, we envisage the applicability of the model in simulating the mechanical response of viscoelastic biological tissues with an anisotropic microstructure, like reproductive tissue.