Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X′π:X→X′ between real algebraic curves where XX is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for XX and for X′X′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X′π:X→X′.