In continuum models for non-perfect fluids, viscoelastic stresses have often been introduced as extra-stresses of purely-dissipative (entropic) nature, similarly to viscous stresses that induce motions of infinite propagation speed. A priori, it requires only one to couple an evolution equation for the (extra-)stress with the momentum balance. In many cases, the apparently-closed resulting system is often not clearly well-posed, even locally in time. The procedure also raises questions about how to encompass transition toward alastic solids. A noticeable exception is K-BZK theory where one starts with a purely elastic fluids. Viscoelasticity then results from dissipative (entropic) stresses due to the relaxation of the fluids'"memory". That K-BKZ approach is physically appealing, but mathematically quite difficult because integrals are introduced to avoid material ('natural') configurations. We propose to introduce viscoelastic stress starting with hyperelastic fluids (like K-BKZ) and evolving material configurations (unlike K-BKZ). At the price of an enlarged system with an additional material-metric variable, one can define well-posed (compressible) motions with finite propagation speed through a system of conservation laws endowed with a "contingent entropy" (like in standard polyconvex elastodynamics).