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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 
$\mathbb {Q}$ and 
$\mathbb {F}_q(t)$, and conclude with a pair of hyperkähler 4-folds over 
$\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.
$K 3$ surfaces
We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general 
$K 3$ surface 
$X$ of degree 
$2$ over 
$ \mathbb{Q} $, together with a 2-torsion Brauer class 
$\alpha $ that is unramified at every finite prime, but ramifies at real points of 
$X$. With motivation from Hodge theory, the pair 
$(X, \alpha )$ is constructed from a double cover of 
${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree 
$(2, 2)$.
