Infrared image super-resolution (IISR) under real-world conditions is a practically significant yet rarely addressed task. Pioneering works are often trained and evaluated on simulated datasets or neglect the intrinsic differences between infrared and visible imaging. In practice, however, real infrared images are affected by coupled optical and sensing degradations that jointly deteriorate both structural sharpness and thermal fidelity. To address these challenges, we propose Real-IISR, a unified autoregressive framework for real-world IISR that progressively reconstructs fine-grained thermal structures and clear backgrounds in a scale-by-scale manner via thermal-structural guided visual autoregression. Specifically, a Thermal-Structural Guidance module encodes thermal priors to mitigate the mismatch between thermal radiation and structural edges. Since non-uniform degradations typically induce quantization bias, Real-IISR adopts a Condition-Adaptive Codebook that dynamically modulates discrete representations based on degradation-aware thermal priors. Also, a Thermal Order Consistency Loss enforces a monotonic relation between temperature and pixel intensity, ensuring relative brightness order rather than absolute values to maintain physical consistency under spatial misalignment and thermal drift. We build FLIR-IISR, a real-world IISR dataset with paired LR-HR infrared images acquired via automated focus variation and motion-induced blur. Extensive experiments demonstrate the promising performance of Real-IISR, providing a unified foundation for real-world IISR and benchmarking. The dataset and code are available at: https://github.com/JZD151/Real-IISR.