  Models of the reionization and reheating of the intergalactic medium (IGM) at
redshifts $z \gtrsim 6$ continue to grow more sophisticated in anticipation of
near-future 21-cm, cosmic microwave background, and galaxy survey measurements.
However, there are many potential sources of systematic uncertainty in models
that could bias and/or degrade upcoming constraints if left unaccounted for. In
this work, we examine three commonly-ignored sources of uncertainty in models
for the mean reionization and thermal histories of the IGM: the underlying
cosmology, halo mass function (HMF), and choice of stellar population synthesis
(SPS) model. We find that cosmological uncertainties affect the Thomson
scattering optical depth at the few percent level and the amplitude of the
global 21-cm signal at the $\sim$5-10 mK level. The differences brought about
by choice of HMF and SPS models are more dramatic, comparable to the $1 \sigma$
error-bar on $\tau_e$ and a $\sim 20$ mK effect on the global 21-cm signal
amplitude. Finally, we jointly fit galaxy luminosity functions and global 21-cm
signals for all HMF/SPS combinations and find that (i) doing so requires
additional free parameters to compensate for modeling systematics and (ii) the
spread in constraints on parameters of interest for different HMF and SPS
choices, assuming $5$ mK noise in the global signal, is comparable to those
obtained when adopting the "true" HMF and SPS with $\gtrsim 20$ mK errors. Our
work highlights the need for dedicated efforts to reduce modeling uncertainties
in order to enable precision inference with future datasets.
