  (Abridged) Combining cosmic shear power spectra and cluster counts is
powerful to improve cosmological parameter constraints and/or test inherent
systematics. However they probe the same cosmic mass density field, if the two
are drawn from the same survey region, and therefore the combination may be
less powerful than first thought. We investigate the cross-covariance between
the cosmic shear power spectra and the cluster counts based on the halo model
approach, where the cross-covariance arises from the three-point correlations
of the underlying mass density field. Fully taking into account the
cross-covariance as well as non-Gaussian errors on the lensing power spectrum
covariance, we find a significant cross-correlation between the lensing power
spectrum signals at multipoles l~10^3 and the cluster counts containing halos
with masses M>10^{14}Msun. Including the cross-covariance for the combined
measurement degrades and in some cases improves the total signal-to-noise
ratios up to plus or minus 20% relative to when the two are independent. For
cosmological parameter determination, the cross-covariance has a smaller effect
as a result of working in a multi-dimensional parameter space, implying that
the two observables can be considered independent to a good approximation. We
also discuss that cluster count experiments using lensing-selected mass peaks
could be more complementary to cosmic shear tomography than mass-selected
cluster counts of the corresponding mass threshold. Using lensing selected
clusters with a realistic usable detection threshold (S/N~6 for a ground-based
survey), the uncertainty on each dark energy parameter may be roughly halved by
the combined experiments, relative to using the power spectra alone.
