We describe a new Monte Carlo (MC) technique that reduces the computational burden of calibration-constrained MC using the concept of the calibration null space. In the new MC approach, the model is calibrated using a subspace regularization method such as Truncated Singular Value Decomposition (TSVD) or the hybrid Tikhonov-TSVD approach described by Tonkin & Doherty (2005). Next, a stochastic parameter field generator is used to produce many realizations of the parameter field. For each realization, a difference is formed between the stochastic field and the calibration field. This difference is projected onto the calibration null space determined through the calibration process, and added to the calibration field. If the model is no longer calibrated, the underlying field is re-estimated with the null-space-difference field "riding on its back". If this can be undertaken using pre-calculated sensitivities, conditioning may require only a very small number of model runs. The new MC approach can rapidly produce a large number of conditioned stochastic fields, for use in assessing the potential error in a wide range of predictions.