This paper presents a novel scan-based method for the real-time monitoring of additive manufacturing processes. Most traditional scanning techniques used for generating 3D models capture the only outer shape of the object after completion of the printing process. The method proposed in this paper differs as it relies on a layer-by-layer scanning of the 3D object directly during the printing process. This strategy has been successfully implemented with a fused filament 3D printer (PRUSA i3 MK3). Furthermore, in order to offer an increased interaction between the obtained 3D model and the user, a virtual environment has been developed for the augmented reality glasses HoloLens. The novelty of this method lies in the layer-by-layer 3D model reconstruction of both the outer shape and the inner layers of the printed part. It enables the user, directly during the printing process, to view and detect potential defects, not only at the surface but also in the inner layers of the printed object. Therefore, it can provide detailed information about the build quality and can be used as the basis of a decision-making tool.