Automating real-time anomaly detection is essential for identifying rare transients in the era of large-scale astronomical surveys. Modern survey telescopes are generating tens of thousands of alerts per night, and future telescopes, such as the Vera C. Rubin Observatory, are projected to increase this number dramatically. Currently, most anomaly detection algorithms for astronomical transients rely either on hand-crafted features extracted from light curves or on features generated through unsupervised representation learning, which are then coupled with standard machine learning anomaly detection algorithms. In this work, we introduce an alternative approach to detecting anomalies: using the penultimate layer of a neural network classifier as the latent space for anomaly detection. We then propose a novel method, named Multi-Class Isolation Forests (MCIF), which trains separate isolation forests for each class to derive an anomaly score for a light curve from the latent space representation given by the classifier. This approach significantly outperforms a standard isolation forest. We also use a simpler input method for real-time transient classifiers which circumvents the need for interpolation in light curves and helps the neural network model inter-passband relationships and handle irregular sampling. Our anomaly detection pipeline identifies rare classes including kilonovae, pair-instability supernovae, and intermediate luminosity transients shortly after trigger on simulated Zwicky Transient Facility light curves. Using a sample of our simulations that matched the population of anomalies expected in nature (54 anomalies and 12,040 common transients), our method was able to discover $41\pm3$ anomalies (~75% recall) after following up the top 2000 (~15%) ranked transients. Our novel method shows that classifiers can be effectively repurposed for real-time anomaly detection.