Description: Deep crustal constraint is often carried out using deterministic inverse methods, sometimes using seismic refraction, gravity and electromagnetic datasets in a complementary or "joint" scheme. With increasingly powerful parallel computer systems it is now possible to apply joint inversion schemes to derive an optimum model from diverse input data. These methods are highly effective where the uncertainty in the system is small. However, given the complex nature of these schemes it is often difficult to discern the uniqueness of the output model given the noise in the data, and the application of necessary regularization and weighting in the inversion process means that the extent of user prejudice pertaining to the final result may be unclear. We can rigorously address the subject of uncertainty using standard statistical tools but these methods also become less feasible if the prior model space is large or the forward simulations are computationally expensive. We present a simple Monte Carlo scheme to screen model space in a fully joint fashion, in which we replace the forward simulation with a fast and uncertainty-calibrated mathematical function, or emulator. This emulator is used as a proxy to run the very large number of models necessary to fully explore the plausible model space. We develop the method using a simple synthetic dataset then demonstrate its use on a joint data set comprising first-arrival seismic refraction. MT and scalar gravity data over a diapiric salt body. This study demonstrates both the value of a forward Monte Carlo approach (as distinct from a search-based or conventional inverse approach) in incorporating all kinds of uncertainty in the modelling process, exploring the entire model space, and shows the potential value of applying emulator technology throughout geophysics. Though the target here is relatively shallow, the methodology can be readily extended to address the whole crust.