We present evidence that the accretion of warm gas onto the Galaxy today is at least as important as cold gas accretion. For more than a decade, the source of the bright Hα emission (up to 750 mR†) along the Magellanic Stream has remained a mystery. We present a hydrodynamical model that explains the known properties of the Hα emission and provides new insights on the lifetime of the Stream clouds. The upstream clouds are gradually disrupted due to their interaction with the hot halo gas. The clouds that follow plough into gas ablated from the upstream clouds, leading to shock ionisation at the leading edges of the downstream clouds. Since the following clouds also experience ablation, and weaker Hα (100–200 mR) is quite extensive, a disruptive cascade must be operating along much of the Stream. In order to light up much of the Stream as observed, it must have a small angle of attack (≈ 20°) to the halo, and this may already find support in new H i observations. Another prediction is that the Balmer ratio (Hα/Hβ) will be substantially enhanced due to the slow shock; this will soon be tested by upcoming WHAM observations in Chile. We find that the clouds are evolving on timescales of 100–200 Myr, such that the Stream must be replenished by the Magellanic Clouds at a fairly constant rate (≳ 0.1 M⊙ yr−1). The ablated material falls onto the Galaxy as a warm drizzle; diffuse ionized gas at 104 K is an important constituent of galactic accretion. The observed Hα emission provides a new constraint on the rate of disruption of the Stream and, consequently, the infall rate of metal-poor gas onto the Galaxy. When the ionized component of the infalling gas is accounted for, the rate of gas accretion is ≳ 0.4 M⊙ yr−1, roughly twice the rate deduced from H i observations alone.