Learning-based image harmonization techniques are usually trained to undo synthetic global transformations, applied to a masked foreground in a single ground truth photo. This simulated data does not model many important appearance mismatches (illumination, object boundaries, etc.) between foreground and background in real composites, leading to models that do not generalize well and cannot model complex local changes. We propose a new semi-supervised training strategy that addresses this problem and lets us learn complex local appearance harmonization from unpaired real composites, where foreground and background come from different images. Our model is fully parametric. It uses RGB curves to correct the global colors and tone and a shading map to model local variations. Our approach outperforms previous work on established benchmarks and real composites, as shown in a user study, and processes high-resolution images interactively. The code and project page is available at https://kewang0622.github.io/sprih/.