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A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace of real elliptic-hyperelliptic algebraic curves in the moduli space of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space of real elliptic-hyperelliptic surfaces of genus > 5: we show that is connected if g is odd and has exactly two connected components if g is even; in both cases the closure of in the compactified moduli space is connected.