  A compact topological surface S, possibly non-orientable and with non-empty
boundary, always admits a Klein surface structure (an atlas whose transition
maps are dianalytic). Its complex cover is, by definition, a compact Riemann
surface M endowed with an anti-holomorphic involution which determines
topologically the original surface S. In this paper, we compare dianalytic
vector bundles over S and holomorphic vector bundles over M, devoting special
attention to the implications that this has for moduli varieties of semistable
vector bundles over M. We construct, starting from S, totally real, totally
geodesic, Lagrangian submanifolds of moduli varieties of semistable vector
bundles of fixed rank and degree over M. This relates the present work to the
constructions of Ho and Liu over non-orientable compact surfaces with empty
boundary (arXiv:math/0605587) .
