Elemental segregation is a ubiquitous phenomenon in additive-manufactured (AM) parts due to solute rejection and redistribution during the solidification process. Using electron microscopy, in situ synchrotron X-ray scattering and diffraction, and thermodynamic modeling, we reveal that in an AM nickel-based superalloy, Inconel 625, stress-relief heat treatment leads to the growth of unwanted δ-phase precipitates on a time scale much faster than that in wrought alloys (minutes versus tens to hundreds of hours). The root cause for this behavior is the elemental segregation that results in local compositions of AM alloys outside the bounds of the allowable range set for wrought alloys. In situ small angle scattering experiments reveal that platelet-shaped δ phase precipitates grow continuously and preferentially along their lateral dimensions during stress-relief heat treatment, while the thickness dimension reaches a plateau very quickly. In situ XRD experiments reveal that nucleation and growth of δ-phase precipitates occur within 5 min during stress-relief heat treatment, indicating a low nucleation barrier and a short incubation time. An activation energy for the growth of δ phase was found to be (131.04 ± 0.69) kJ mol-1. We further demonstrate that a subsequent homogenization heat treatment can effectively homogenize the AM alloy and remove the deleterious δ phase. The combined experimental and modeling methodology in this work can be extended to elucidate the phase evolution during heat treatments in a broad range of AM materials.