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 semistable 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.