  The purpose of this paper is to compute determinant index bundles of certain
families of Real Dirac type operators on Klein surfaces as elements in the
corresponding Grothendieck group of Real line bundles in the sense of Atiyah.
On a Klein surface these determinant index bundles have a natural holomorphic
description as theta line bundles. In particular we compute the first
Stiefel-Whitney classes of the corresponding fixed point bundles on the real
part of the Picard torus. The computation of these classes is important,
because they control to a large extent the orientability of certain moduli
spaces in Real gauge theory and Real algebraic geometry.
