When a posterior peaks in unexpected regions of parameter space, new physics has either been discovered, or a bias has not been identified yet. To tell these two cases apart is of paramount importance. We therefore present a hybrid between likelihood-based analyses and forward-modelling, with the aim of indicating—and mitigating—unrecognized biases: our method runs any pipeline with possibly unknown biases on both simulations and real data. It computes the coverage probability of posteriors, which measures whether posterior volume is a faithful representation of probability or not. If found to be necessary, the posterior is then corrected. This is a non-parametric debiasing procedure which complies with objective Bayesian inference. We exemplify the method by debiasing inference with approximate covariance matrices and redshift uncertainties. We demonstrate why approximate covariance matrices bias physical constraints, and how this bias can be mitigated. We show that for a Euclid-like survey, if a traditional likelihood exists, then 25 end-to-end simulations suffice to guarantee that the figure of merit deteriorates maximally by 22 percent, or by 10 percent for 225 simulations. Thus, even a pessimistic analysis of Euclid-like data will still constitute a 25-fold increase in precision on the dark energy parameters in comparison to the state of the art (2018) set by KiDS and DES. Our code is publicly available.