The complex interaction between shocks and plasma turbulence is extremely important to address crucial features of energy conversion in a broad range of astrophysical systems. We study the interaction between a supercritical, perpendicular shock and pre-existing, fully developed plasma turbulence, employing a novel combination of magnetohydrodynamic and small-scale, hybrid-kinetic simulations where a shock is propagating through a turbulent medium. The variability of the shock front in the unperturbed case and for two levels of upstream fluctuations is addressed. We find that the behaviour of shock ripples, i.e. shock surface fluctuations with short (a few ion skin depths, di) wavelengths, is modified by the presence of pre-existing turbulence, which also induces strong corrugations of the shock front at larger scales. We link this complex behaviour of the shock front and the shock downstream structuring with the proton temperature anisotropies produced in the shock-turbulence system. Finally, we put our modelling effort in the context of spacecraft observations, elucidating the role of novel cross-scale, multispacecraft measurements in resolving shock front irregularities at different scales. These results are relevant for a broad range of astrophysical systems characterized by the presence of shock waves interacting with plasma turbulence.