  We give the first examples of derived equivalences between varieties defined
over non-closed fields where one has a rational point and the other does not.
We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude
with a pair of hyperkaehler 4-folds over Q. The latter is independently
interesting as a new example of a transcendental Brauer-Manin obstruction to
the Hasse principle.
