Prestressed stayed columns offer an innovative and aesthetically attractive solution where designs demand long and slender elements under compression. The addition of cross-arms and pretensioned cables to the columns provides significant lateral and rotational support that can result in a marked increase of the buckling capacity. The current work presents a simplified model, comprising discrete rigid links and springs with a mathematically exact formulation including the destabilising effect of cable slackening, which mimics the response of stayed columns and reveals the governing physical parameters that drive the behaviour. A recent finite-element study had indicated that these structural components are likely to exhibit interactive buckling phenomena, particularly if the critical buckling mode is antisymmetric, leading to highly unstable responses. However, by its very nature, that study could not reveal the bifurcational structure of the equilibrium response. The results from the present model reveal similar behaviour to that observed in the earlier finite-element study. A sequence of bifurcations is found that switch the post-buckling response from one mode to another by way of a path of interactive buckling.