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(Abridged) Combining cosmic shear power spectra and cluster counts ispowerful to improve cosmological parameter constraints and/or test inherentsystematics. However they probe the same cosmic mass density field, if the twoare drawn from the same survey region, and therefore the combination may beless powerful than first thought. We investigate the cross-covariance betweenthe cosmic shear power spectra and the cluster counts based on the halo modelapproach, where the cross-covariance arises from the three-point correlationsof the underlying mass density field. Fully taking into account thecross-covariance as well as non-Gaussian errors on the lensing power spectrumcovariance, we find a significant cross-correlation between the lensing powerspectrum signals at multipoles l~10^3 and the cluster counts containing haloswith masses M>10^{14}Msun. Including the cross-covariance for the combinedmeasurement degrades and in some cases improves the total signal-to-noiseratios up to plus or minus 20% relative to when the two are independent. Forcosmological parameter determination, the cross-covariance has a smaller effectas a result of working in a multi-dimensional parameter space, implying thatthe two observables can be considered independent to a good approximation. Wealso discuss that cluster count experiments using lensing-selected mass peakscould be more complementary to cosmic shear tomography than mass-selectedcluster counts of the corresponding mass threshold. Using lensing selectedclusters with a realistic usable detection threshold (S/N~6 for a ground-basedsurvey), the uncertainty on each dark energy parameter may be roughly halved bythe combined experiments, relative to using the power spectra alone.

Probing dark energy with cluster counts and cosmic shear power spectra: including the full covariance