  We study arithmetic properties of del Pezzo surfaces of degree 4 for which
the Brauer group has the largest possible order using different fibrations into
curves. We show that if such a surface admits a conic fibration, then it always
has a rational point. We also answer a question of V\'arilly-Alvarado and Viray
by showing that the Brauer groups these surfaces cannot be vertical with
respect to any projection away from a plane. We conclude that the available
techniques for proving existence of rational points or even Zariski density do
not directly apply if there is no Brauer-Manin obstruction to the Hasse
principle.
  In passing we pick up the first examples of quartic del Pezzo surfaces with a
Brauer group of order 4 for which the failure of the Hasse principle is
explained by a Brauer-Manin obstruction.
