Additively manufacturing (AM) of Ti-6Al-4V alloy via laser powder-bed fusion (LPBF) results in a non-equilibrium martensitic (α’) microstructure, with high strength but poor ductility and toughness. These properties may be modified by post-AM heat treatment, in which the α’ phase decomposes into an equilibrium α+β structure, while conserving some microstructural features (e.g. length scales) of the α’ lath structure. Here, we combine experimental and computational methods to explore the thermodynamics and kinetics of martensitic decomposition in LPBF-produced Ti-6Al-4V alloy. Experiments rely on state-of-the-art in-situ characterization by electron microscopy imaging and diffraction during multi-step heat treatment between 400˚C and the alloy β-transus temperature (995˚C). Computational simulations rely on a simple, yet experimentally-informed phase-field model, solved using a spectral Fourier-based method parallelized on graphical processing units. Experiments confirmed that as-built microstructures were fully composed of martensitic α’ laths and that nucleation of the β phase occurs primarily along α’ lath boundaries, with traces of β nucleation sites along crystalline defects. Corresponding phase-field simulations directly use experimentally-assessed EBSD maps of as-printed microstructures and the experimental thermal history, and simulation results are compared directly with in-situ characterisation data. Experiments and simulations confirm that, while full decomposition into stable α+β phases was already complete at 700˚C, visible morphological evolution of the microstructure was only observed for T≥700˚C, without seen any change of the prior β grain structure during the heat treatment.