Double covers of smooth quadric threefolds with Artin–Mumford obstructions to rationality

We study obstructions to rationality on a nodal Fano threefold $M$ that is a double cover of a smooth quadric threefold ramified over an intersection with a quartic threefold in $P^4$. We prove that if $M$ admits an Artin–Mumford obstruction to rationality then it lies in one of three explicitly described families. Conversely, a general element of any of these families admits an Artin–Mumford obstruction to rationality. Only one of these three families was known before; other two families of nodal Fano threefolds with obstructions to rationality are new.