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The cosmological information contained in anisotropic galaxy clustering measurements can often be compressed into a small number of parameters whose posterior distribution is well described by a Gaussian. We present a general methodology to combine these estimates into a single set of consensus constraints that encode the total information of the individual measurements, taking into account the full covariance between the different methods. We illustrate this technique by applying it to combine the results obtained from different clustering analyses, including measurements of the signature of baryon acoustic oscillations (BAO) and redshift-space distortions (RSD), based on a set of mock catalogues of the final SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). Our results show that the region of the parameter space allowed by the consensus constraints is smaller than that of the individual methods, highlighting the importance of performing multiple analyses on galaxy surveys even when the measurements are highly correlated. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The methodology presented here is used in Alam et al. (2016) to produce the final cosmological constraints from BOSS.