  We examine the existence of rational divisors on modular curves of
$\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves
over local fields. Using a criterion of Poonen and Stoll, we show that in
infinitely many cases the Tate-Shafarevich groups of the Jacobians of these
Atkin-Lehner quotients have non-square orders.
