Implementations of Markov chain Monte Carlo (MCMC) methods need to confront two fundamental challenges: accurate representation of prior information and efficient evaluation of likelihood functions. The definition and sampling of the prior distribution can often be facilitated by standard dimensionality-reduction techniques such as Principal Component Analysis (PCA). Additionally, PCA-based decompositions can enable the implementation of accurate surrogate models, for instance, based on polynomial chaos expansion (PCE). However, intricate geological priors with sharp contrasts may demand advanced dimensionality-reduction techniques, such as deep generative models (DGMs). Although suitable for prior sampling, these DGMs pose challenges for surrogate modelling. In this contribution, we present a MCMC strategy that combines the high reconstruction performance of a DGM in the form of a variational autoencoder with the accuracy of PCA–PCE surrogate modelling. Additionally, we introduce a physics-informed PCA decomposition to improve accuracy and reduce the computational burden associated with surrogate modelling. Our methodology is exemplified in the context of Bayesian ground-penetrating radar traveltime tomography using channelized subsurface structures, providing accurate reconstructions and significant speed-ups, particularly when the computation of the full-physics forward model is costly.