  (ABRIDGED) We present a Far-Ultraviolet Spectroscopic Explorer survey of
highly ionized high-velocity clouds (HVCs) in 66 extragalactic sight lines. We
find a total of 63 high-velocity O VI absorbers, 16 with 21 cm-emitting H I
counterparts and 47 ``highly ionized'' absorbers without 21 cm emission. 11 of
these high-velocity O VI absorbers are positive-velocity wings (broad O VI
features extending asymmetrically to velocities of up to 300 km/s). The highly
ionized HVC population is characterized by <b(O VI)>=38+/-10 km/s and <log
N_a(O VI)>=13.83+/-0.36. We find that 81% (30/37) of high-velocity O VI
absorbers have clear accompanying C III absorption, and 76% (29/38) have
accompanying H I absorption in the Lyman series. 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. We find that the shape of the wing profiles is
well reproduced by a radiatively cooling, vertical outflow. However, the
outflow has to be patchy and out of ionization equilibrium. An alternative
model, consistent with the observations, 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. Though we cannot rule out a Local
Group explanation, 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 non-detection 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.
