  We develop estimators of agreement and disagreement between correlated
cosmological data sets. These account for data correlations when computing the
significance of both tensions and excess confirmation while remaining
statistically optimal. We discuss and thoroughly characterize different
approaches commenting on the ones that have the best behavior in practical
applications. We complement the calculation of their statistical distribution
within the Gaussian model with one estimator that takes non-Gaussianities fully
into account. To illustrate the use of our techniques, we apply these
estimators to supernovae measurements of the distance-redshift relation,
absolutely calibrated by the local distance ladder. The suite of best
estimators that we discuss finds results that are in excellent agreement
between estimators and find no indications of significant internal
inconsistencies in this data set above the $1\%$ probability threshold. This
shows the robustness of local determinations of the Hubble constant to features
in the distance-redshift relation.
