Tandem solar cells consisting of high bandgap cadmium telluride alloys atop crystalline silicon have potential for high efficiencies exceeding the Shockley-Queisser limit. However, experimental results have fallen well below this goal significantly because of non-ideal current matching and light trapping. In this work, we simulate cadmium zinc telluride (CZT) and crystalline silicon (c-Si) tandems as an exemplary system to show the role that a hybrid light trapping and bandgap engineering approach can play in improving performance and lowering materials costs for tandem solar cells incorporating crystalline silicon. This work consists of two steps. First, we optimize absorption in the crystalline silicon layer with front pyramidal texturing and asymmetric dielectric back gratings, which results in 121% absorption enhancement from a planar structure. Then, using this pre-optimized light trapping scheme, we model the dispersion of the CdxZn1-xTe alloys, and then adjust the bandgap to realize the best current matching for a range of CZT thicknesses. Using experimental parameters, the corresponding maximum efficiency is predicted to be 16.08 % for a total tandem cell thickness of only 2.2 μm.