Volcanic eruption models are hampered by the lack of multiphase magmatic flow laws. Most rheological models estimate the viscosity of multiphase lavas via the Einstein-Roscoe equation, but this simplification cannot be used for high crystallinity and it does not consider the non-Newtonian strain-rate dependence of viscosity. We carried out parallel plate experiments on natural samples to simulate multiphase lava deformation under various stresses and strain rates. Multiphase lavas exhibit an important component of shear thinning, and appear to invalidate the adequacy of Einstein-Roscoe–based formulations for highly crystalline lava rheology. The remarkable singular dependence of viscosity (η) on strain rate (γ) yields a novel universal rheology law at eruptive temperatures (T), i.e., log η = −0.993 + 8974/T −0.543·log γ Our work reveals the importance of considering microcracking and viscous dissipation at very high strain rate (>10−3 s−1), explaining the occurrence of seismic swarms along the conduit margins, and consequently supporting plug-like magma ascent models.