  We report on the measurement of the beam spin asymmetry in the deeply virtual
Compton scattering off $^4$He using the CEBAF Large Acceptance Spectrometer
(CLAS) at Jefferson Lab using a 6 GeV longitudinally polarized electron beam
incident on a pressurized $^4$He gaseous target. We detail the method used to
ensure the exclusivity of the measured reactions, in particular the upgrade of
CLAS with a radial time projection chamber to detect the low-energy recoiling
$^4$He nuclei and an inner calorimeter to extend the photon detection
acceptance at forward angles. Our results confirm the theoretically predicted
enhancement of the coherent ($e^4$He$~\to~e'$$^4$He$'\gamma'$) beam spin
asymmetries compared to those observed on the free proton, while the incoherent
($e^4$He$~\to~e'$p$'\gamma'$X$'$) asymmetries exhibit a 30$\%$ suppression.
From the coherent data, we were able to extract, in a model-independent way,
the real and imaginary parts of the only $^4$He Compton form factor, $\cal
H_A$, leading the way toward 3D imaging of the partonic structure of nuclei.
