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We consider equations of Müller-Israel-Stewart type describing a relativistic viscous fluid with bulk viscosity in four-dimensional Minkowski space. We show that there exists a class of smooth initial data that are localized perturbations of constant states for which the corresponding unique solutions to the Cauchy problem break down in finite time. Specifically, we prove that in finite time such solutions develop a singularity or become unphysical in a sense that we make precise. We also show that in general Riemann invariants do not exist in 1+1 dimensions for physically relevant equations of state and viscosity coefficients. Finally, we present a more general version of a result by Y. Guo and A.S. Tahvildar-Zadeh: we prove large-data singularity formation results for perfect fluids under very general assumptions on the equation of state, allowing any value for the fluid sound speed strictly less than the speed of light.