Full-shell nanowires (a semiconducting core fully wrapped by an epitaxial superconducting shell) have recently been introduced as promising hybrid quantum devices. Despite this, however, their properties when forming a Josephson junction (JJ) have not been elucidated yet. We here fill this void by theoretically studying the physics of JJs based on full-shell nanowires. In the hollow-core limit, where the thickness of the semiconducting layer can be ignored, we demonstrate that the critical supercurrent $I^{c}$ can be tuned by an external magnetic flux $Φ$. Specifically, $I^{c}(Φ)$ does not follow the Little-Parks modulation of the superconducting pairing $Δ(Φ)$, and exhibits steps for realistic values of nanowire radii. The position of the steps can be understood from the underlying symmetries of the orbital transverse channels which contribute to the supercurrent for a given chemical potential.

12 pages