Tori and surfaces violating a local-to-global principle for rationality

We show that even within a class of varieties where the Brauer–Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field k each of those, in the absence of the Brauer obstruction to rationality, is rational over all completions of k but is not k-rational.