A 150-km-wide ground deformation anomaly in the Altiplano-Puna volcanic complex (APVC) of the Central Andes, with uplift centered on Uturuncu volcano and peripheral subsidence, alludes to complex subsurface stress changes. In particular, the role of a large, geophysically anomalous and partially molten reservoir (the Altiplano-Puna magma body, APMB), located ~20 km beneath the deforming surface, is still poorly understood. To explain the observed spatiotemporal ground deformation pattern, we integrate geophysical and petrological data and develop a numerical model that accounts for a mechanically heterogeneous and viscoelastic crust. Best-fit models imply subsurface stress changes due to the episodic reorganization of an interconnected vertically extended mid-crustal plumbing system composed of the APMB and a domed bulge and column structure. Measured gravity-height gradient data point toward low-density fluid migration as the dominant process behind these stress changes. We calculate a mean annual flux of ~2 × 10 7 m 3 of water-rich andesitic melt and/or magmatic water from the APMB into the bulge and column structure accompanied by modest pressure changes of <0.006 MPa/yr. Two configurations of the column fit the observations equally well: (1) a magmatic (igneous mush) column that extends to a depth of 6 km below sea level and contains trapped volatiles, or (2) a volatile-bearing hybrid column composed of an igneous mush below a solidified and permeable body that extends to sea level. Volatile loss from the bulge and column structure reverses the deformation, and explains the absence of broad (tens of kilometers) and long-term (>100 yr) residual deformation at Uturuncu. Episodic mush reorganization may be a ubiquitous characteristic of the magmatic evolution of the APVC.