  We show that transcendental elements of the Brauer group of an algebraic
surface can obstruct the Hasse principle. We construct a general K3 surface X
of degree 2 over Q, together with a two-torsion Brauer class A that is
unramified at every finite prime, but ramifies at real points of X. Motivated
by Hodge theory, the pair (X,A) is constructed from a double cover of P^2 x P^2
ramified over a hypersurface of bi-degree (2,2).
