The layered transition metal dichalcogenides which show an inherent tendency towards charge density wave order are traditionally considered to realize rather two-dimensional electronic systems. However, recent theoretical and experimental results suggest that the stacking of the charge density wave and related orbital order in the direction perpendicular to the layers plays a key role for the in-plane electronic structure. Here we present a state-of-the-art density functional theory (DFT) study which models crucial features of the partially disordered orbital order stacking in the prototypical layered transition metal dichalcogenide 1T-TaS2. We show that a DFT model with realistic assumptions about the orbital order perpendicular to the layers yields band structures which agree remarkably well with experiments and also correctly predicts the formation of an excitation gap at the Fermi energy. Our results not only imply that the widely accepted paradigm of local Mott physics as the driving mechanism behind the gap formation in 1T-TaS2 needs to be reconsidered but they also highlight the crucial role of interlayer interactions in layered transition metal dichalcogenides in general.