We propose a novel approach for solving inverse-problems with high-dimensional inputs and an expensive forward mapping. It leverages joint deep generative modelling to transfer the original problem spaces to a lower dimensional latent space. By jointly modelling input and output variables and endowing the latent with a prior distribution, the fitted probabilistic model indirectly gives access to the approximate conditional distributions of interest. Since model error and observational noise with unknown distributions are common in practice, we resort to likelihood-free inference with Approximate Bayesian Computation (ABC). Our method calls on ABC by Subset Simulation to explore the regions of the latent space with dissimilarities between generated and observed outputs below prescribed thresholds. We diagnose the diversity of approximate posterior solutions by monitoring the probability content of these regions as a function of the threshold. We further analyze the curvature of the resulting diagnostic curve to propose an adequate ABC threshold. When applied to a cross-borehole tomography example from geophysics, our approach delivers promising performance without using prior knowledge of the forward nor of the noise distribution.