  Galaxy intrinsic alignments (IAs) have long been recognised as a significant
contaminant to weak lensing-based cosmological inference. In this paper we seek
to quantify the impact of a common modelling assumption in analytic
descriptions of intrinsic alignments: that of spherically symmetric dark matter
halos. Understanding such effects is important as the current generation of
intrinsic alignment models are known to be limited, particularly on small
scales, and building an accurate theoretical description will be essential for
fully exploiting the information in future lensing data. Our analysis is based
on a catalogue of 113,560 galaxies between $z=0.06-1.00$ from MassiveBlack-II,
a hydrodynamical simulation of box length $100 h^{-1}$ Mpc. We find satellite
anisotropy contributes at the level of $\geq 30-40\%$ to the small scale
alignment correlation functions. At separations larger than $1 h^{-1}$ Mpc the
impact is roughly scale-independent, inducing a shift in the amplitude of the
IA power spectra of $\sim20\%$. These conclusions are consistent across the
redshift range and between the MassiveBlack-II and Illustris simulations. The
cosmological implications of these results are tested using a simulated
likelihood analysis. Synthetic cosmic shear data is constructed with the
expected characteristics (depth, area and number density) of a future LSST-like
survey. Our results suggest that modelling alignments using a halo model based
upon spherical symmetry could potentially induce cosmological parameter biases
at the $\sim 1.5\sigma$ level for $S_8$ and $w$.
