  We present MUSE/VLT imaging spectroscopy of rest-frame ultraviolet emission
lines and ALMA observations of the [CI] $^3$P$_1$-$^3$P$_0$ emission line,
probing both the ionized and diffuse molecular medium around the radio galaxy
4C 19.71 at z~3.6. This radio galaxy has extended Ly$\alpha$ emission over a
region ~100 kpc in size preferentially oriented along the axis of the radio
jet. Faint Ly$\alpha$ emission extends beyond the radio hot spots. We also find
extended CIV and HeII emission over a region of ~150 kpc in size, where the
most distant emission lies ~40 kpc beyond the north radio lobe and has narrow
FWHM line widths of ~180 km/s and a small relative velocity offset
$\Delta$v~130 km/s from the systemic redshift of the radio galaxy. The [CI] is
detected in the same region with FWHM~100 km/s and $\Delta$v~5 km/s, while [CI]
is not detected in the regions south of the radio galaxy. We interpret the
coincidence in the northern line emission as evidence of relatively quiescent
multi-phase gas residing within the halo at a projected distance of ~75 kpc
from the host galaxy. To test this hypothesis, we performed photoionization and
photo-dissociated region (PDR) modeling, using the code Cloudy, of the three
emission line regions: the radio galaxy proper and the northern and southern
regions. We find that the [CI]/CIV$\lambda\lambda$1548,1551 and
CIV$\lambda\lambda$1548,1551/HeII ratios of the two halo regions are consistent
with a PDR or ionization front in the circumgalactic medium likely energized by
photons from the active galactic nuclei. This modeling is consistent with a
relatively low metallicity, 0.03<[Z/Z$_{\odot}$]<0.1, and diffuse ionization
with an ionization parameter (proportional to the ratio of the photon number
density and gas density) of log U~3 for the two circumgalactic line emission
regions. Using rough mass estimates for the molecular [...]
