Today's dominant photovoltaic technologies rely on single-junction devices, which are approaching their practical efficiency limit of 25-27%. Therefore, researchers are increasingly turning to multi-junction devices, which consist of two or more stacked subcells, each absorbing a different part of the solar spectrum. Here, we show that dual-junction III-V//Si devices with mechanically stacked, independently operated III-V and Si cells reach cumulative one-sun efficiencies up to 32.8%. Efficiencies up to 35.9% were achieved when combining a GaInP/GaAs dual-junction cell with a Si single-junction cell. These efficiencies exceed both the theoretical 29.4% efficiency limit of conventional Si technology and the efficiency of the record III-V dual-junction device (32.6%), highlighting the potential of Si-based multi-junction solar cells. However, techno-economic analysis reveals an order-of-magnitude disparity between the costs for III-V//Si tandem cells and conventional Si solar cells, which can be reduced if research advances in low-cost III-V growth techniques and new substrate materials are successful.