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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/$α$, and, in two cases, ionizing flux fields that are softer and brighter than the fiducial Haardt & Madau UV background model.