We study dynamic heterogeneities in a model glass-former whose overlap with a reference configuration is constrained to a fixed value. The system phase-separates into regions of small and large overlap, so that dynamical correlations remain strong even for asymptotic times. We calculate an appropriate thermodynamic potential and find evidence of a Maxwell's construction consistent with a spinodal decomposition of two phases. Our results suggest that dynamic heterogeneities are the expression of an ephemeral phase-separating regime ruled by a finite surface tension.The conspicuous lack of a growing correlation length, contrasting with the very steep increase of the relaxation time, has been a puzzle in the physics of structural glasses for quite a long time. Arguably, the first breakthrough has been the discovery of dynamic heterogeneities [1], and the detection of a growing dynamical correlation length, ξ d [2,3]. If we take two snapshots of the system separated by a time lag comparable to the α relaxation time, τ α , the particle displacements vary enormously across the system, and the typical size ξ d of the mobility-correlated regions increases on lowering the temperature.More recently, by studying the thermodynamics of systems subject to amorphous boundary conditions [4,5], an entirely different, fully static, correlation length ξ s has been discovered [6,7]. ξ s also grows upon cooling, even though its surge occurs at lower temperatures than ξ d . The static correlation length has a natural interpretation as the size of the cooperatively rearranging regions [8], and within the random first-order theory [9] it is determined by the balance between a surface tension cost and a configurational entropy gain of a rearrangement.Clearly, it would be desirable to unify the dynamic and the thermodynamic frameworks, so as to understand the interplay between the two correlation lengths. Although the static-dynamic connection is clear in mean-field systems [10] and some progresses have been made in more realistic systems [11], we are quite far from a unifying picture in real glass-formers. Here we show that surface tension, which is a crucial ingredient of the thermodynamic framework, also plays a key role in the formation * Present address: CEA, Institut de Physique Theorique, Saclay, F-91191 Gif-sur-Yvette, France † Present address: SMC-INFM and Dipartimento di Fisica, Univesitá di Roma "Sapienza", p.le Aldo Moro 5, 00185, Roma, Italy. of dynamic heterogeneities. In so doing, we establish a much-needed further link between dynamic and thermodynamic relaxation in glass-forming liquids.Let us start with the standard measurement of the dynamic correlation length ξ d . Our glass-former is the well-known soft-sphere model in 3-d [26]. A useful tool to measure ξ d is the overlap, which quantifies how much a configuration at time t is similar to the reference configuration at t = 0. If we partition the system in small cubic boxes and let n i be the number of particles in box i, the local overlap is defined as q(r i , t) ≡ n ...