Rational points on twisted K3 surfaces and derived equivalences

Using a construction of Hassett–V'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, ${\mathbb{Q}}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.