  We investigate how the choice of equation of state (EOS) and resolution
conspire to affect the outcomes of giant impact (GI) simulations. We focus on
the simple case of equal mass collisions of two Earth-like $0.5\,M_\oplus$
proto-planets showing that the choice of EOS has a profound impact on the
outcome of such collisions as well as on the numerical convergence with
resolution. In simulations where the Tillotson EOS is used, impacts generate an
excess amount of vapour due to the lack of a thermodynamically consistent
treatment of phase transitions and mixtures. In oblique collisions this
enhances the artificial angular momentum (AM) transport from the planet to the
circum-planetary disc reducing the planet's rotation period over time. Even at
a resolution of $1.3 \times 10^6$ particles the result is not converged. In
head-on collisions the lack of a proper treatment of the solid/liquid-vapour
phase transition allows the bound material to expand to very low densities
which in turn results in very slow numerical convergence of the critical
specific impact energy for catastrophic disruption $Q_{RD}^*$ with increasing
resolution as reported in prior work. The simulations where ANEOS is used for
oblique impacts are already converged at a modest resolution of $10^5$
particles, while head-on collisions converge when they evidence the post-shock
formation of a dense iron-rich ring, which promotes gravitational
re-accumulation of material. Once sufficient resolution is reached to resolve
the liquid-vapour phase transition of iron in the ANEOS case, and this ring is
resolved, the value of $Q_{RD}^*$ has then converged.
