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