A threefold violating a local-to-global principle for rationality

In this note we construct an example of a smooth projective threefold that is irrational over \(\mathbb {Q}\) $\mathbb{Q}$ but is rational at all places. Our example is a complete intersection of two quadrics in \(\mathbb P^5\) ${\mathbb{P}}^5$ , and we show it has the desired rationality behavior by constructing an explicit element of order \(4\) $4$ in the Tate–Shafarevich group of the Jacobian of an associated genus \(2\) $2$ curve.