Heterogeneous computers combine a general-purpose host processor with domain-specific programmable many-core accelerators, uniting high versatility with high performance and energy efficiency. While the host manages ever-more application memory, accelerators are designed to work mainly on their local memory. This difference in addressed memory leads to a discrepancy between the optimal address width of the host and the accelerator. Today 64-bit host processors are commonplace, but few accelerators exceed 32-bit address-able local memory, a difference expected to increase with 128-bit hosts in the exascale era. Managing this discrepancy requires support for multiple data models in heterogeneous compilers. So far, compiler support for multiple data models has not been explored, which hampers the programmability of such systems and inhibits their adoption.In this work, we perform the first exploration of the feasibility and performance of implementing a mixed-data-model heterogeneous system. To support this, we present and evaluate the first mixed-data-model compiler, supporting arbitrary address widths on host and accelerator. To hide the inherent complexity and to enable high programmer productivity, we implement transparent offloading on top of OpenMP. The proposed compiler techniques are implemented in LLVM and evaluated on a 64+32-bit heterogeneous SoC. Results on benchmarks from the PolyBench-ACC suite show that memory can be transparently shared between host and accelerator at overheads below 0.7% compared to 32-bit-only execution, enabling mixed-data-model computers to execute at near-native performance.