Combining a Si solar cell with a high-bandgap top cell reduces the thermalization losses in the short wavelength and enables theoretical 1-sun efficiencies far over 30%. We have investigated the fabrication and optimization of Si-based tandem solar cells with 1.8-eV rear-heterojunction GaInP top cells. The III-V and Si heterojunction subcells were fabricated separately and joined by mechanical stacking using electrically insulating optically transparent interlayers. Our GaInP/Si dual-junction solar cells have achieved a certified cumulative 1-sun efficiency of 29.8% +/- 0.6% (AM1.5g) in four-terminal operation conditions, which exceeds the record 1-sun efficiencies achieved with both III-V and Si single-junction solar cells. The effect of luminescent coupling between the subcells has been investigated, and optical losses in the solar cell structure have been addressed.