We investigate correlations induced by gravitational lensing on simulated cosmic microwave background data of experiments with an incomplete sky coverage and their effect on inferences from the South Pole Telescope (SPT) data. These correlations agree well with the theoretical expectations, given by the sum of supersample and intrasample lensing terms, with only a typically negligible ∼5 % discrepancy in the amplitude of the supersample lensing effect. Including these effects we find that lensing constraints are in 3.0 σ or 2.1 σ tension between the SPT polarization measurements and Planck temperature or lensing reconstruction constraints respectively. If the lensing-induced covariance effects are neglected, the significance of these tensions increases to 3.5 σ or 2.5 σ . Using the standard scaling parameter AL substantially underestimates the significance of the tension once other parameters are marginalized over. By parameterizing the supersample lensing through the mean convergence in the SPT footprint, we find a hint of underdensity in the SPT region. We also constrain extra sharpening of the cosmic microwave background acoustic peaks due to missing smoothing of the peaks by supersample lenses at a level that is much smaller than the lens sample variance. Finally, we extend the usual "shift in the means" statistic for evaluating tensions to non-Gaussian posteriors, generalize an approach to extract correlation modes from noisy simulated covariance matrices, and present a treatment of correlation modes not as data covariances but as auxiliary model parameters.