  Cross-correlations of CMB lensing reconstructions with other tracers of
matter constrain primordial non-Gaussianity, neutrino masses and structure
growth as a function of cosmic time. We formalize a method to improve the
precision of these measurements by using a third tracer to remove structure
from the lensing reconstructions. Crucially, our method enjoys the variance
reduction benefits of a joint-modelling approach without the need to model the
cosmological dependence of the ancillary tracer. We present a first
demonstration of variance cancellation using data from Planck and the DESI
Legacy Surveys, showing a 10-20% reduction in both lensing power and
cross-correlation variance using the Cosmic Infrared Background (CIB) or DESI
Legacy Survey Luminous Red Galaxies (LRGs) as matter tracers.
