  The cross-correlation between 21-cm intensity mapping experiments and
photometric surveys of galaxies (or any other cosmological tracer with a broad
radial kernel) is severely degraded by the loss of long-wavelength radial modes
due to Galactic foreground contamination. Higher-order correlators are able to
restore some of these modes due to the non-linear coupling between them and the
local small-scale clustering induced by gravitational collapse. We explore the
possibility of recovering information from the bispectrum between a photometric
galaxy sample and an intensity mapping experiment, in the context of the
clustering-redshifts technique. We demonstrate that the bispectrum is able to
calibrate the redshift distribution of the photometric sample to the required
accuracy of future experiments such as the Rubin Observatory, using future
single-dish and interferometric 21-cm observations, in situations where the
two-point function is not able to do so due to foreground contamination. We
also show how this calibration is affected by the photometric redshift width
$\sigma_{z,0}$ and maximum scale $k_{\mathrm{max}}$. We find that it is
important to reach scales $k \gtrsim 0.3\,h\,\mathrm{Mpc}^{-1}$, with the
constraints saturating at around $k\sim 1\,h\,\mathrm{Mpc}^{-1}$ for
next-generation experiments.
