  We show that odd order transcendental elements of the Brauer group of a K3
surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$
of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class
$\alpha$ that is unramified at all primes except for 3, but ramifies at all
3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha)$ is
constructed from a cubic fourfold $X$ of discriminant 18 birational to a
fibration into sextic del Pezzo surfaces over the projective plane. Notably,
our construction does not rely on the presence of a central simple algebra
representative for $\alpha$. Instead, we prove that a sufficient condition for
such a Brauer class to obstruct the Hasse principle is insolubility of the
fourfold $X$ (and hence the fibers) over $\mathbb{Q}_3$ and local solubility at
all other primes.
