We present a new model for the hadronisation of multi-parton systems, in which colour correlations beyond leading are allowed to influence the formation of confining potentials (strings). The multiplet structure of is combined with a minimisation of the string potential energy, to decide between which partons strings should form, allowing also for "baryonic" configurations (e.g., two colours can combine coherently to form an anticolour). In collisions, modifications to the leading-colour picture are small, suppressed by both colour and kinematics factors. But in collisions, multi-parton interactions increase the number of possible subleading connections, counteracting their naive suppression. Moreover, those that reduce the overall string lengths are kinematically favoured. The model, which we have implemented in the PYTHIA 8 generator, is capable of reaching agreement not only with the important distribution but also with measured rates (and ratios) of kaons and hyperons, in both and collisions. Nonetheless, the shape of their spectra remains challenging to explain.