Symmetric positive-definite system solvers based on Cholesky factorization are fundamental to many scientific applications, such as climate modeling. We present a portable, nested recursive mixed-precision solver designed for Matrix Processing Units (MXUs), including NVIDIA Tensor Cores (H200) and AMD Matrix Cores (MI300X), that assigns low-precision FP16 arithmetic to large off-diagonal blocks, while preserving high precision on diagonal blocks to ensure numerical stability. The solver is implemented in Julia, providing a high-level, hardware-agnostic interface. We demonstrate up to a 5.07x speedup relative to the diagonal-precision vendor baseline, with 100x better accuracy than pure half precision on H200, providing higher accuracy than low-precision at higher speed than high-precision. Positive performance trends are also observed on MI300X, demonstrating broad applicability across GPUs.