The phase diagrams and the nature of the phase transitions in multicomponent gauge theories with an Abelian gauge field are important topics with various physical applications. While an early renormalization-group-based study indicated that the direct transition from a fully ordered to a fully disordered state is continuous for N=1 and N>183, recently it was demonstrated that the transition is discontinuous for N=2. We quantitatively study the dependence on N of the degree of discontinuity of this transition. Our results suggest that the transition is discontinuous at least up to N=7. Furthermore, we demonstrate that, at increased coupling strength, the phase transitions of the neutral and charged sectors of the model split, which for N>2 yields a new phase with composite order. The transition from the composite-order phase to the fully disordered phase is then also discontinuous, at least for N=3 and N=4. Via a duality argument, this indicates that van der Waals-type interaction between directed loops may be responsible for the discontinuous phase transitions in these models.