  Given a geometrically irreducible smooth projective curve of genus 1 defined
over the field of real numbers, and a pair of integers r and d, we determine
the isomorphism class of the moduli space of semi-stable vector bundles of rank
r and degree d on the curve. When r and d are coprime, we describe the topology
of the real locus and give a modular interpretation of its points. We also
study, for arbitrary rank and degree, the moduli space of indecomposable vector
bundles of rank r and degree d, and determine its isomorphism class as a real
algebraic variety.
