  Dynamic interactions between the two Magellanic Clouds have flung large
quantities of gas into the halo of the Milky Way, creating the Magellanic
Stream, the Magellanic Bridge, and the Leading Arm (collectively referred to as
the Magellanic System). In this third paper of a series studying the Magellanic
gas in absorption, we analyze the gas ionization level using a sample of 69
Hubble Space Telescope/Cosmic Origins Spectrograph sightlines that pass through
or within 30 degrees of the 21 cm-emitting regions. We find that 81% (56/69) of
the sightlines show UV absorption at Magellanic velocities, indicating that the
total cross section of the Magellanic System is ~11 000 square degrees, or
around a quarter of the entire sky. Using observations of the Si III/Si II
ratio together with Cloudy photoionization modeling, we calculate that the
total mass (atomic plus ionized) of the Magellanic System is ~2.0 billion solar
masses, with the ionized gas contributing over twice as much mass as the atomic
gas. This is larger than the current-day interstellar H I mass of both
Magellanic Clouds combined, indicating that they have lost most of their
initial gas mass. If the gas in the Magellanic System survives to reach the
Galactic disk over its inflow time of ~0.5-1.5 Gyr, it will represent an
average inflow rate of ~3.7-6.7 solar masses per year, potentially raising the
Galactic star formation rate. However, multiple signs of an evaporative
interaction with the hot Galactic corona indicate that the Stream may not
survive its journey to the disk fully intact, and will instead add material to
(and cool) the corona.
