Interaction is one of the core abilities of humanoid robots. However, most existing frameworks focus on non-interactive whole-body control, which limits their practical applicability. In this work, we develop InterReal, a unified physics-based imitation learning framework for Real-world human-object Interaction (HOI) control. InterReal enables humanoid robots to track HOI reference motions, facilitating the learning of fine-grained interactive skills and their deployment in real-world settings. Within this framework, we first introduce a HOI motion data augmentation scheme with hand-object contact constraints, and utilize the augmented motions to improve policy stability under object perturbations. Second, we propose an automatic reward learner to address the challenge of large-scale reward shaping. A meta-policy guided by critical tracking error metrics explores and allocates reward signals to the low-level reinforcement learning objective, which enables more effective learning of interactive policies. Experiments on HOI tasks of box-picking and box-pushing demonstrate that InterReal achieves the best tracking accuracy and the highest task success rate compared to recent baselines. Furthermore, we validate the framework on the real-world robot Unitree G1, which demonstrates its practical effectiveness and robustness beyond simulation.