  Almost one decade ago, Poonen constructed the first examples of algebraic
varieties over global fields for which Skorobogatov's etale Brauer-Manin
obstruction does not explain the failure of the Hasse principle. By now,
several constructions are known, but they all share common geometric features
such as large fundamental groups.
  In this paper, we construct simply connected fourfolds over global fields of
positive characteristic for which the Brauer-Manin machinery fails. Contrary to
earlier work in this direction, our construction does not rely on major
conjectures. Instead, we establish a new diophantine result of independent
interest: a Mordell-type theorem for Campana's "geometric orbifolds" over
function fields of positive characteristic. Along the way, we also construct
the first example of simply connected surface of general type over a global
field with a non-empty, but non-Zariski dense set of rational points.
