The Magellanic Stream is sculpted by its infall through the Milky Way's circumgalactic medium, but the rates and directions of mass, momentum, and energy exchange through the Stream-halo interface are relative unknowns critical for determining the origin and fate of the Stream. Complementary to large-scale simulations of LMC-SMC interactions, we apply new insights derived from idealized, highresolution "cloud-crushing" and radiative turbulent mixing layer simulations to the Leading Arm and Trailing Stream. Contrary to classical expectations of fast cloud breakup, we predict that the Leading Arm and much of the Trailing Stream should be surviving infall and even gaining mass due to strong radiative cooling. Provided a sufficiently supersonic tidal swing-out from the Clouds, the present-day Leading Arm could be a series of high-density clumps in the cooling tail behind the progenitor cloud. We back up our analytic framework with a suite of converged wind-tunnel simulations, finding that previous results on cloud survival and mass growth can be extended to high Mach number (M) flows with a modified drag time t drag ∝ 1 + M and longer growth time. We also simulate the Trailing Stream; we find that the growth time is long (∼ Gyrs) compared to the infall time, and approximate Hα emission is low on average (∼few mR) but can be up to tens of mR in bright spots. Our findings also have broader extragalactic implications for e.g. galactic winds, which we discuss.