  Within integrable systems, the class of so called "semitoric" integrable
systems in dimension four has attracted a lot of attention in recent years,
especially since fundamental examples from classical and quantum mechanics have
been identified as semitoric by different groups of researchers. Several of
these examples, however, show a particular trait not included in the original
theory, that is, the presence of multiple (i.e. two or more) rank zero isolated
singularities in the same energy-momentum level sets. Systems with this
property are called non-simple. This paper extends the original theory of
Pelayo and V\~u Ngoc to non-simple systems.
