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.
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