In a previous paper, we analyzed a class of time travel paradoxes which cannot be resolved using Novikov's self-consistency conjecture, meaning that the system is fundamentally and irreparably inconsistent. We proved that the paradoxes can nonetheless always be resolved, and the system made consistent, by assuming that traveling back in time creates a new independent history (or timeline), such that any changes to the past affect only the new history and not the original one. Therefore, we argued that if time travel is possible, that would necessarily imply the existence of multiple histories. However, our proof was obtained using a simplistic and unrealistic toy model, which was formulated using contrived laws of physics. The purpose of the present paper is to define and analyze a new model of time travel paradoxes, which is fully compatible with all known physics - provided, of course, that time travel itself is possible. This model consists of a wormhole time machine in 3+1 spacetime dimensions, which can be either permanent (existing eternally) or temporary (activated only for a short time). We define the spacetime topology and geometry of the model, calculate the geodesics of objects passing through the time machine, and prove that this model inevitably leads to paradoxes which cannot be resolved using Novikov's conjecture, but can be resolved using multiple histories. This result provides more substantial support to our claim that time travel necessarily implies multiple histories.