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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> each of those, in the absence of the Brauer obstruction to rationality, is rational over all completions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> but is not <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -rational.