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Mapping facies is very desirable for hydrocarbon field planning and forecasting. Yet inversion of true-amplitude imaged seismic data for rock and fluid lithotypes is a difficult inverse problem, even with convivial assumptions about the character of migrated seismic data. Since seismic data typically contain no information in certain frequency bands, Bayesian or equivalent priors must be introduced, and this missing information is partly supplied by the discrete spatial distribution of facies. The need to work with facies, or mixtures of rock-property distributions, induces unpleasant nonconvexity in the inversion objective. This raises at least two major challenges: (i) sensitivity of any inversions to model (parameter) misspecification in the Bayesian prior distributions, and (ii) adequate globally-optimal inversions from the models thus specified. We use a Markov random field for the Bayesian facies prior. Parameter inference for these models is a known NP-hard problem, but good approximations using belief propagation answer the first challenge. Approaches to problem (ii) involve removing the elastic parameters analytically, and using specially crafted global optimisation techniques on the remaining space of discrete facies variables. Two fruitful approaches are annealing, and semidefinite programming relaxations. The latter produce remarkably tight convex relaxations, bettered only by carefully tuned annealing scheme.