Normal coverings of hyperelliptic real algebraic curves

We consider normal (possibly) branched, finite-sheeted coverings $\pi:X\rightarrow X'$ between hyperelliptic real algebraic curves. 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 in case $X$ has the maximum number of ovals within its genus. We first analyze the topological features and ramification data of such coverings. For each isomorphism class we then describe a representative, with defining polynomial equations for $X$ and for $X'$, formulae for generators of the covering transformation group, and a rational formula for the covering $\pi:X\rightarrow X'$.