When doing a $2$-descent on the Jacobian $J$ of a curve of genus $2$, one wishes to determine whether or not certain twists of $J$ have rational points. As $J$ and its twists are unwieldy, we consider the quotient $K$ of a twist by the involution induced by multiplication by $-1$ on $J$. We construct an explicit curve $C$ and corresponding twist for which there is a Brauer-Manin obstruction to the existence of rational points on $K$. This yields infinitely many twists of $C$ with nontrivial Tate-Shafarevich group. This is joint work with Adam Logan at Waterloo.