  We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a
$(2,2)$-divisor, which have the structure of a conic bundle threefold with
smooth quartic discriminant curve via the second projection. In each isotopy
class of smooth plane quartics, we construct examples where the total space of
the conic bundle is rational. For five of the six isotopy classes we construct
$\mathbb C$-rational examples that have obstructions to rationality over
$\mathbb R$, and for the sixth class, we show that the models we consider are
all rational. Moreover, for three of the five classes with irrational members,
we give characterizations of rationality using the topology of the real locus
and the intermediate Jacobian torsor obstruction of Hassett--Tschinkel and
Benoist--Wittenberg. The double cover models we consider were introduced and
previously studied by S. Frei, S. Sankar, B. Viray, I. Vogt, and the first
author.
