We propose a new system that efficiently combines direct multitouch interaction and 3D stereoscopic visualization. In our approach, users interact by way of simple 2D gestures on a monoscopic touchscreen, while visualizing occlusion-free 3D stereoscopic objects floating above the surface at an optically correct distance. By registering the 3D virtual space with the physical space, we produce a rich seamless workspace where both the advantages of direct and indirect interaction are jointly exploited. In addition to standard multitouch gestures and controls (e.g. pan, zoom, and standard 2D widgets) from which we take advantage, we have designed a dedicated multitouch 3D transformation widget. This widget allows the near-direct control of rotations, scaling, and translations of the manipulated objects. To illustrate the power of our setup, we have designed a demo scenario where participants reassemble 3D virtual fragments. This scenario, as many others, takes benefit of our proposal, where the strength of both multitouch interaction and stereoscopic visualization are unified in an innovative and relevant workspace.