We propose a sophisticated framework for high-energy hadronic collisions, wherein different QCD physics processes are interleaved in a common sequence of falling transverse-momentum values. Thereby phase-space competition is introduced between multiple parton-parton interactions and initialstate radiation. As a first step we develop new transverse-momentum-ordered showers for initial- and final-state radiation, which should be of use also beyond the scope of the current article. These showers are then applied in the context of multiple interactions, and a few tests of the new model are presented. The article concludes with an outlook on further aspects, such as the possibility of a shower branching giving partons participating in two different interactions.