The assembly of biopolymers into a hydrated elastic network often goes along with syneresis, a spontaneous process during which the hydrogel slowly shrinks and releases solvent. The tendency to syneresis of calcium-alginate hydrogels, widely used biocompatible materials, is a hindrance to applications for which dimensional integrity is crucial. Although calcium-induced aggregation of specific block-sequences has been long known as the microscopic process at work in both primary cross-linking and syneresis, the nature of the coupling between these structural events and the global deswelling flow has remained so far elusive. We have tackled this issue within the regime of entangled pregels that yield highly cross-linked, self-crowded hydrogels with stiff networks. Using an original, stopped-flow extrusion experiment, we have unveiled a robust, stretched-exponential kinetics of shrinking, spanning more than six decades of time and quasi-independent of the alginate concentration. A careful analysis of the puzzling dynamical features of syneresis in these gels has led us to propose that due to the network rigidity, the calcium-fueled, random collapse events that drive solvent locally, are not thermally activated but rather controlled by the average poroelastic flow itself, according to a self-sustained mechanism described here for the first time.