  One major goal of upcoming large-scale-structure surveys is to constrain dark
energy and modified gravity theories. In particular, galaxy clustering and
gravitational lensing convergence are probes sensitive to modifications of
general relativity. While the standard analysis for these surveys typically
includes power spectra or 2-point correlation functions, it is known that the
bispectrum contains more information, and could offer improved constraints on
parameters when combined with the power spectra. However, the use of bispectra
has been limited so far to one single probe, e.g. the lensing convergence
bispectrum or the galaxy bispectrum. In this paper, we extend the formalism to
explore the power of cross-bispectra between different probes, and exploit
their ability to break parameter degeneracies and improve constraints. We study
this on a test case of lensing convergence and galaxy density auto- and
cross-bispectra, for a particular sub-class of Horndeski theories parametrized
by $c_M$ and $c_B$. Using the 2000 deg$^2$ notional survey of the Nancy Grace
Roman Space Telescope with overlapping photometry from the Rubin Observatory
Legacy Survey of Space and Time, we find that a joint power spectra and
bispectra analysis with three redshift bins at $l_{\rm max} = 1000$ yields
$\sigma_{c_M} = 1.0$ and $\sigma_{c_B} = 0.3$, both a factor of $\sim$1.2
better than the power spectra results; this would be further improved to
$\sigma_{c_M} = 0.7$ and $\sigma_{c_B} = 0.2$ if $l_{\rm max} = 3000$ is taken.
Furthermore, we find that using all possible cross-bispectra between the two
probes in different tomographic bins improves upon auto-bispectra results by a
factor of 1.3 in $\sigma_{c_M}$, 1.1 in $\sigma_{c_B}$ and 1.3 in
$\sigma_{\Omega_m}$. We expect that similar benefits of using cross-bispectra
between probes could apply to other science cases and surveys.
