Lensing-ISW bispectrum-related cross spectra computed from Eq. (46). Black dashed lines indicate the average value for simulations, while dark/light grey filled regions indicate the expected one/two standard deviation scatter, also measured from simulations. The thin magenta line gives the expected cross-spectrum for our fiducial model. The agreement of this curve with the simulation average illustrates that our estimator is accurately normalized. In all the quantitative analysis of this section we ignore L< 10, although we have plotted the cross-spectra at these multipoles for interest.