  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 data. These
correlations agree well with the theoretical expectations, given by the sum of
super-sample and intra-sample lensing terms, with only a typically negligible
$\sim$ 5% discrepancy in the amplitude of the super-sample lensing effect.
Including these effects we find that lensing constraints are in $3.0\sigma$ or
$2.1\sigma$ 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 $3.5\sigma$ or $2.5\sigma$. Using the standard scaling
parameter $A_L$ substantially underestimates the significance of the tension
once other parameters are marginalized over. By parameterizing the super-sample
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 CMB
acoustic peaks due to missing smoothing of the peaks by super-sample 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.
