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Markov chain Monte Carlo (MCMC) methods have become standard in Bayesian inference and multi‐observable inversions in almost every discipline of the Earth sciences. In the case of geodynamic and/or coupled geophysical‐geodynamic inverse problems, however, the computational cost associated with the solution of large‐scale 3‐D Stokes forward problems has rendered probabilistic formulations impractical. Here we present a novel and extremely efficient method to produce ultrafast solutions of the 3‐D Stokes problem for MCMC simulations. Our approach combines the individual benefits of Reduced Basis techniques, goal‐oriented error formulations, and MCMC algorithms to produce an accurate and computationally efficient surrogate for the forward problem. Importantly, the surrogate adapts itself during the MCMC simulation according to the history of the chain and the goals of the inversion. This maximizes the efficiency of the forward problem and removes the need for preinversion off‐line computations to build a surrogate. We demonstrate the benefits and limitations of the method with several numerical examples and show that in all cases the computational cost is of the order of <1% compared to a traditional MCMC approach. The method is general enough to be applied to a range of problems, including uncertainty quantification/propagation, adjoint‐based geodynamic inversions, sensitivity analyses in mantle convection problems, and in the creating surrogate models for complex forward problems (e.g., heat transfer, seismic tomography, and magnetotellurics).