  The pursuit of experimental precision in the $CP$-violating weak phase
$\phi_2$ ($\alpha$) is not without its challenges, in part due to the need to
combine multiple physical observables from various related decay channels, and
therein lies a fundamental issue. Similarities in analysis procedures give rise
to systematic correlations between the measured inputs constraining $\phi_2$
that must be taken into account to avoid bias. Specifically, in the case of the
irreducible model uncertainty accompanying analyses involving the $\rho$ meson,
it is demonstrated that ignoring correlations derived from its pole parameters,
or indeed even treating correlations individually contained within each decay
channel, can ultimately lead to a bias in $\phi_2$ of $\mathcal{O}(1^\circ)$.
Correct treatment on the other hand, markedly reduces wandering of its central
value as a function of the model uncertainty strength with the added dividend
of a further improved overall uncertainty. Bias in the combination of $B^0 \to
(\rho \pi)^0$ and $B \to \rho \rho$ is also seen to depend on the statistical
strength of the former in relation to that of the model uncertainty in the
latter. This work can inspire other studies into the points at which systematic
correlations beyond those determined in single measurements matter in
combinations leading to other $CP$-violating weak phases such as $\phi_1$
($\beta$), $\phi_3$ ($\gamma$) and $\phi_s$.
