  In 2011, V\`arilly-Alvarado and the last author constructed an Enriques
surface $X$ over $\mathbb{Q}$ with an \'etale-Brauer obstruction to the Hasse
principle and no algebraic Brauer-Manin obstruction. In this paper, we show
that the nontrivial Brauer class of $X_{\bar{\mathbb{Q}}}$ does not descend to
$\mathbb{Q}$. Together with the results of V\`arilly-Alvarado and the last
author, this proves that the Brauer-Manin obstruction is insufficient to
explain all failures of the Hasse principle on Enriques surfaces. The methods
of this paper build on the ideas in several recent papers by the last author
and various collaborators: we study geometrically unramified Brauer classes on
$X$ via pullback of ramified Brauer classes on a rational surface. Notably, we
develop techniques which work over fields which are not necessarily separably
closed, in particular, over number fields.
