Infrared Image Super-Resolution (IRSR) is challenged by the low contrast and sparse textures of infrared data, requiring robust long-range modeling to maintain global coherence. While State-Space Models like Mamba offer proficiency in modeling long-range dependencies for this task, their inherent 1D causal scanning mechanism fragments the global context of 2D images, hindering fine-detail restoration. To address this, we propose Global Phase and Spectral Prompt-guided Mamba (GPSMamba), a framework that synergizes architectural guidance with non-causal supervision. First, our Adaptive Semantic-Frequency State Space Module (ASF-SSM) injects a fused semantic-frequency prompt directly into the Mamba block, integrating non-local context to guide reconstruction. Then, a novel Thermal-Spectral Attention and Phase Consistency Loss provides explicit, non-causal supervision to enforce global structural and spectral fidelity. By combining these two innovations, our work presents a systematic strategy to mitigate the limitations of causal modeling. Extensive experiments demonstrate that GPSMamba achieves state-of-the-art performance, validating our approach as a powerful new paradigm for infrared image restoration. Code is available at https://github.com/yongsongH/GPSMamba.