  Quasar absorption systems encode a wealth of information about the
abundances, ionization structure, and physical conditions in intergalactic and
circumgalactic media. Simple (often single-phase) photoionization models are
frequently used to decode such data. Using five discrete absorbers from the COS
Absorption Survey of Baryon Harbors (CASBaH) that exhibit a wide range of
detected ions (e.g., Mg II, S II--S VI, O II--O VI, Ne VIII), we show several
examples where single-phase ionization models cannot reproduce the full set of
measured column densities. To explore models that can self-consistently explain
the measurements and kinematic alignment of disparate ions, we develop a
Bayesian multiphase ionization modeling framework that characterizes discrete
phases by their unique physical conditions and also investigates variations in
the shape of the UV flux field, metallicity, and relative abundances. Our
models require at least two (but favor three) distinct ionization phases
ranging from $T \approx 10^{4}$ K photoionized gas to warm-hot phases at $T
\lesssim 10^{5.8}$ K. For some ions, an apparently single absorption
"component" includes contributions from more than one phase, and up to 30% of
the H I is not from the lowest ionization phase. If we assume that all of the
phases are photoionized, we cannot find solutions in thermal pressure
equilibrium. By introducing hotter, collisionally ionized phases, however, we
can achieve balanced pressures. The best models indicate moderate
metallicities, often with sub-solar N/$\alpha$, and, in two cases, ionizing
flux fields that are softer and brighter than the fiducial Haardt & Madau UV
background model.

Astrophysics of Galaxies