  We present a new method for joint cosmological parameter inference and
cluster mass calibration from a combination of weak lensing measurements and
the abundance of thermal Sunyaev-Zel'dovich (tSZ) selected galaxy clusters. We
combine cluster counts with the spherical harmonic cosmic shear power spectrum
and the cross-correlation between cluster overdensity and cosmic shear. These
correlations constrain the cluster mass-observable relation. We model the
observables using a halo model framework, including their full non-Gaussian
covariance. Forecasting constraints on cosmological and mass calibration
parameters for a combination of LSST cosmic shear and Simons Observatory tSZ
cluster counts, we find competitive constraints for cluster cosmology, with a
factor two improvement in the dark energy figure of merit compared to LSST
cosmic shear alone. We find most of the mass calibration information will be in
the large and intermediate scales of the cross-correlation between cluster
overdensity and cosmic shear. Finally, we find broadly comparable constraints
to traditional analyses based on calibrating masses using stacked cluster
lensing measurements, with the benefit of consistently accounting for the
correlations with cosmic shear.
