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New supernova surveys such as the Dark Energy Survey, Pan-STARRS and the LSST will produce an unprecedented number of photometric supernova candidates, most with no spectroscopic data. Avoiding biases in cosmological parameters due to the resulting inevitable contamination from non-Ia supernovae can be achieved with the BEAMS formalism, allowing for fully photometric supernova cosmology studies. Here we extend BEAMS to deal with the case in which the supernovae are correlated by systematic uncertainties. The analytical form of the full BEAMS posterior requires evaluating 2 N terms, where N is the number of supernova candidates. Thisexponential catastrophe'is computationally unfeasible even for N of order 100. We circumvent the exponential catastrophe by marginalising numerically instead of analytically over the possible supernova types: we augment the cosmological parameters with …