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The dominant gaseous structure in the Galactic halo is the Magellanic Stream, an extended network of neutral and ionized filaments surrounding the Large and Small Magellanic Clouds (LMC/SMC), the two most massive satellite galaxies of the Milky Way. Recent observations indicate that the Clouds are on their first passage around our Galaxy, the Stream is made up of gas stripped from both the LMC and the SMC, and the majority of this gas is ionized. While it has long been suspected that tidal forces and ram-pressure stripping contributed to the Stream's formation, a full understanding of its origins has defied modelers for decades. Several recent developments, including the discovery of dwarf galaxies associated with the Magellanic Group, the high mass of the LMC, the detection of highly ionized gas toward stars in the LMC and the predictions of cosmological simulations all support the existence of a halo of warm ionized gas around the LMC at a temperature of $\sim5\times10^{5}\;\mathrm{K}$. Here we show that by including this "Magellanic Corona" in hydrodynamic simulations of the Magellanic Clouds falling onto the Galaxy, we can simultaneously reproduce the Stream and its Leading Arm. Our simulations explain the Stream's filamentary structure, spatial extent, radial velocity gradient, and total ionized gas mass. We predict that the Magellanic Corona will be unambiguously observable via high-ionization absorption lines in the ultraviolet spectra of background quasars lying near the LMC. This prediction is directly testable with the Cosmic Origins Spectrograph on the Hubble Space Telescope.