The Magellanic Stream and Leading Arm of H I that stretches from the Large and Small Magellanic Clouds (LMC and SMC) and over 200° of the Southern sky is thought to be formed from multiple encounters between the LMC and SMC. In this scenario, most of the gas in the Stream and Leading Arm is stripped from the SMC, yet recent observations have shown a bifurcation of the Trailing Arm that reveals LMC origins for some of the gas. Absorption measurements in the Stream also reveal an order of magnitude more gas than in current tidal models. We present hydrodynamical simulations of the multiple encounters between the LMC and SMC at their first pass around the Milky Way, assuming that the Clouds were more extended and gas-rich in the past. Our models create filamentary structures of gas in the Trailing Stream from both the LMC and SMC. While the SMC trailing filament matches the observed Stream location, the LMC filament is offset. In addition, the total observed mass of the Stream in these models is underestimated by a factor of four when the ionized component is accounted for. Our results suggest that there should also be gas stripped from both the LMC and SMC in the Leading Arm, mirroring the bifurcation in the Trailing Stream. This prediction is consistent with recent measurements of spatial variation in chemical abundances in the Leading Arm, which show that gas from multiple sources is present, although its nature is still uncertain.