The adaptive quantised tracking problem is addressed for a class of switched stochastic strict-feedback non-linear systems with asymmetrical input saturation in this study. With the aid of a Gaussian error function-based continuous differentiable switching model and some special techniques, the technical difficulties from dealing with the switching asymmetric saturation non-linearities and the sector-bounded quantisation errors are overcome. Then, by combining the common Lyapunov function method, backstepping technique and neural network approximation-based approach, a simple common adaptive tracking control scheme involving one adaptive parameter only is presented for such systems under arbitrary switching. The given quantised control scheme guarantees that all signals of the closed-loop system are semi-globally bounded in probability while the tracking error can converge to a small neighbourhood of the origin. Finally, simulation studies are provided to illustrate the effectiveness and applicability of the proposed control design.