  We investigate various phenomenological schemes for the rapid generation of
3D mock galaxy catalogues with a given power spectrum and bispectrum. We apply
the fast bispectrum estimator \MODALLSS{} to these mock galaxy catalogues and
compare to $N$-body simulation data analysed with the halo-finder
\texttt{ROCKSTAR} (our benchmark data). We propose an assembly bias model for
populating parent halos with subhalos by using a joint lognormal-Gaussian
probability distribution for the subhalo occupation number and the halo
concentration. This prescription enabled us to recover the benchmark power
spectrum from $N$-body simulations to within 1\% and the bispectrum to within
4\% across the entire range of scales of the simulation. A small further boost
adding an extra galaxy to all parent halos above the mass threshold
$M>2\times10^{14}\,h^{-1} M_\odot$ obtained a better than 1\% fit to both power
spectrum and bispectrum in the range $K/3<1.1\,h\,\text{Mpc}^{-1}$, where
$K=k_1+k_2+k_3$. This statistical model should be applicable to fast dark
matter codes, allowing rapid generation of mock catalogues which simultaneously
reproduce the halo power spectrum and bispectrum obtained from $N$-body
simulations. We also investigate alternative schemes using the Halo Occupation
Distribution (HOD) which depend only on halo mass, but these yield results
deficient in both the power spectrum (2\%) and the bispectrum (>4\%) at $k,K/3
\approx 0.2\,h\,\text{Mpc}^{-1}$, with poor scaling for the latter. Efforts to
match the power spectrum by modifying the standard four-parameter HOD model
result in overboosting the bispectrum (with a 10\% excess). We also
characterise the effect of changing the halo profile on the power spectrum and
bispectrum.
