In this work, we study the problem of object re-identification (ReID) in a 3D multi-object tracking (MOT) context, by learning to match pairs of objects from cropped (e.g., using their predicted 3D bounding boxes) point cloud observations. We are not concerned with SOTA performance for 3D MOT, however. Instead, we seek to answer the following question: In a realistic tracking by-detection context, how does object ReID from point clouds perform relative to ReID from images? To enable such a study, we propose a lightweight matching head that can be concatenated to any set or sequence processing backbone (e.g., PointNet or ViT), creating a family of comparable object ReID networks for both modalities. Run in siamese style, our proposed point-cloud ReID networks can make thousands of pairwise comparisons in real-time (10 hz). Our findings demonstrate that their performance increases with higher sensor resolution and approaches that of image ReID when observations are sufficiently dense. Additionally, we investigate our network's ability to enhance 3D multi-object tracking (MOT), showing that our point-cloud ReID networks can successfully re-identify objects which led a strong motion-based tracker into error. To our knowledge, we are the first to study real-time object re-identification from point clouds in a 3D multi-object tracking context.