Linguists have traditionally represented patterns of divergence within a language family in terms of either a 'splits' model, corresponding to a branching family tree structure, or the wave model, resulting in a (dialect) continuum. Recent phylogenetic analyses, however, have tended to assume the former as a viable idealization also for the latter. But the contrast matters, for it typically reflects different processes in the real world: speaker populations either separated by migrations, or expanding over continuous territory. Since history often leaves a complex of both patterns within the same language family, ideally we need a single model to capture both, and tease apart the respective contributions of each. The 'network' type of phylogenetic method offers this, so we review recent applications to language data. Most have used lexical data, encoded as binary or multi-state characters. We look instead at continuous distance measures of divergence in phonetics. Our output networks combine branch- and continuum-like signals in ways that correspond well to known histories (illustrated for Germanic, and particularly English). We thus challenge the traditional insistence on shared innovations, setting out a new, principled explanation for why complex language histories can emerge correctly from distance measures, despite shared retentions and parallel innovations.