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We present a Far Ultraviolet Spectroscopic Explorer survey of highly ionized high-velocity clouds (HVCs) in 66 extragalactic sight lines with (S/N)₁₀₃₀>8. We search the spectra for high-velocity (100 km/s<|v_LSR|<400 km/s) O VI absorption and find a total of 63 absorbers, 16 with 21 cm emitting H I counterparts and 47 ``highly ionized'' absorbers without 21 cm emission. The highly ionized HVC population is characterized by <b(O VI)≥38±10 km/s and <logN<sub>a</sub>(O VI)≥13.83±0.36, with negative-velocity clouds generally found at l<180° and positive-velocity clouds found at l>180°. Eleven of these highly ionized HVCs are positive-velocity wings (broad O VI features extending asymmetrically to velocities of up to 300 km/s). We find that 81% (30 of 37) of highly ionized HVCs have clear accompanying C III absorption, and 76% (29 of 38) have accompanying H I absorption in the Lyman series. We present the first (O VI selected) sample of C III and H I absorption line HVCs and find <b(C III)≥30±8 km/s, logN<sub>a</sub>(C III) ranges from <12.5 to >14.4, <b(H I)≥22±5 km/s, and log N<sub>a</sub>(H I) ranges from <14.7 to >16.9. The lower average width of the high-velocity H I absorbers implies the H I lines arise in a separate, lower temperature phase than the O VI. The ratio N<sub>a</sub>(C III)/N<sub>a</sub>(O VI) is generally constant with velocity in highly ionized HVCs, suggesting that at least some C III resides in the same gas as the O VI. Collisional ionization equilibrium models with solar abundances can explain the O VI/C III ratios for temperatures near 1.7x10<sup>5</sup> K; nonequilibrium models with the O VI ``frozen in'' at lower temperatures are also possible. Photoionization models are not viable since they underpredict O VI by several orders of magnitude. The presence of associated C III and H I strongly suggests the highly ionized HVCs are not formed in the hotter plasma that gives rise to O VII and O VIII X-ray absorption. We find that the shape of the O VI positive-velocity wing profiles is well reproduced by a radiatively cooling, vertical outflow moving with ballistic dynamics, with T₀=10⁶ K, n₀~2x10^–3/cm3, and v₀~250 km/s. However, the outflow has to be patchy and out of ionization equilibrium to explain the sky distribution and the simultaneous presence of O VI, C III, and H I. We found that a spherical outflow can produce high-velocity O VI components (as opposed to the wings), showing that the possible range of outflow model results is too broad to conclusively identify whether or not an outflow has left its signature in the data. An alternative model, supported by the similar multiphase structure and similar O VI properties of highly ionized and 21 cm HVCs, is one where the highly ionized HVCs represent the low N(H I) tail of the HVC population, with the O VI formed at the interfaces around the embedded H I cores. Although we cannot rule out the possibility that some highly ionized HVCs exist in the Local Group or beyond, we favor a Galactic origin. This is based on the recent evidence that both H I HVCs and the million-degree gas detected in X-ray absorption are Galactic phenomena. Since the highly ionized HVCs appear to trace the interface between these two Galactic phases, it follows that highly ionized HVCs are Galactic themselves. However, the nondetection of high-velocity O VI in halo star spectra implies that any Galactic high-velocity O VI exists at z distances beyond a few kpc.