  We study when the period and the index of a class in the Brauer group of the
function field of a real algebraic surface coincide. We prove that it is always
the case if the surface has no real points (more generally, if the class
vanishes in restriction to the real points of the locus where it is
well-defined), and give a necessary and sufficient condition for unramified
classes. As an application, we show that the u-invariant of the function field
of a real algebraic surface is equal to 4, answering questions of Lang and
Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jong's
period-index theorem on complex surfaces.
