We present FACT, a method for predicting alignment quality (i.e., registration error) of registered lidar point cloud pairs. This is useful e.g. for quality assurance of large, automatically registered 3D models. FACT extracts local features from a registered pair and processes them with a point transformer-based network to predict a misalignment class. We generalize prior work that study binary alignment classification of registration errors, by recasting it as multinomial misalignment classification. To achieve this, we introduce a custom regression-by-classification loss function that combines the cross-entropy and Wasserstein losses, and demonstrate that it outperforms both direct regression and prior binary classification. FACT successfully classifies point-cloud pairs registered with both the classical ICP and GeoTransformer, while other choices, such as standard point-cloud-quality metrics and registration residuals are shown to be poor choices for predicting misalignment. On a synthetically perturbed point-cloud task introduced by the CorAl method, we show that FACT achieves substantially better performance than CorAl. Finally, we demonstrate how FACT can assist experts in correcting misaligned point-cloud maps. Our code is available at https://github.com/LudvigDillen/FACT_for_PCMC .