  A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a
general $(2,2)$-divisor will have the structure of a geometrically standard
conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$
via the second projection. These threefolds are rational over algebraically
closed fields, but over nonclosed fields, including over $\mathbb{R}$, their
rationality is an open problem. In this paper, we characterize rationality over
$\mathbb{R}$ when $\Delta(\mathbb{R})$ has at least two connected components
(extending work of M. Ji and the second author) and over local fields when all
odd degree fibers of the first projection have nonsquare discriminant.
  We obtain these applications by proving general results comparing the conic
bundle structure on $Y$ with the conic bundle structure on a well-chosen
intersection of two quadrics. The difference between these two conic bundles is
encoded by a constant Brauer class, and we prove that this class measures a
certain failure of Galois descent for the codimension 2 Chow group of $Y$.
