  Utilizing Planck polarized dust emission maps at 353 GHz and new large-area
maps of the neutral hydrogen (HI) cold neutral medium (CNM) fraction
($f_\mathrm{CNM}$), we investigate the relationship between dust polarization
fraction ($p_{353}$) and $f_\mathrm{CNM}$ in the diffuse high latitude
($|b|>30^{\circ}$) sky. We find that the correlation between $p_{353}$ and
$f_\mathrm{CNM}$ is qualitatively distinct from the $p_{353}$-HI column density
($N_\mathrm{H\,I}$) relationship. At low column densities
($N_\mathrm{H\,I}<4\times10^{20}~\mathrm{cm^{-2}}$) where $p_{353}$ and
$N_\mathrm{H\,I}$ are uncorrelated, there is a strong positive
$p_{353}$-$f_\mathrm{CNM}$ correlation. We fit the $p_{353}$-$f_{\rm CNM}$
correlation with data-driven models to constrain the degree of magnetic field
disorder between phases along the line-of-sight. We argue that an increased
magnetic field disorder in the warm neutral medium (WNM) relative to the CNM
best explains the positive $p_{353}$-$f_\mathrm{CNM}$ correlation in diffuse
regions. Modeling the CNM-associated dust column as being maximally polarized,
with a polarization fraction $p_{\rm CNM} \sim$ 0.2, we find that the best-fit
mean polarization fraction in the WNM-associated dust column is 0.22$p_{\rm
CNM}$. The model further suggests that a significant $f_{\rm CNM}$-correlated
fraction of the non-CNM column (an additional ~18.4% of the HI mass on average)
is also more magnetically ordered, and we speculate that the additional column
is associated with the unstable medium (UNM). Our results constitute a new
large-area constraint on the average relative disorder of magnetic fields
between the neutral phases of the ISM, and are consistent with the physical
picture of a more magnetically aligned CNM column forming out of a disordered
WNM.
