Quartic del Pezzo surfaces with a Brauer group of order 4

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.