Conic bundle threefolds differing by a constant Brauer class and connections to rationality

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; however, over nonclosed fields, including $\mathbb{R}$, 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 encodes the obstruction to the existence of a section of the first projection $Y\to\mathbb{P}^1$.