This article presents a pioneering approach to real-time spacecraft pose estimation, utilizing a mixed-precision quantized neural network implemented on the FPGA components of a commercially available Xilinx MPSoC, renowned for its suitability in space applications. Our co-design methodology includes a novel evaluation technique for assessing the layer-wise neural network sensitivity to quantization, facilitating an optimal balance between accuracy, latency, and FPGA resource utilization. Utilizing the FINN library, we developed a bespoke FPGA dataflow accelerator that integrates on-chip weights and activation functions to minimize latency and energy consumption. Our implementation is 7.7 times faster and 19.5 times more energy-efficient than the best-reported values in the existing spacecraft pose estimation literature. Furthermore, our contribution includes the first real-time, open-source implementation of such algorithms, marking a significant advancement in making efficient spacecraft pose estimation algorithms widely accessible. The source code is available at https://github.com/possoj/FPGA-SpacePose.