Laparoscopic surgery has evolved as a key technique for cancer diagnosis and therapy. While characterization of the tissue perfusion is crucial in various procedures, such as partial nephrectomy, doing so by means of visual inspection remains highly challenging. Spectral imaging takes advantage of the fact that different tissue components have unique optical properties to recover relevant information on tissue function such as ischemia. However, clinical success stories for advancing laparoscopic surgery with spectral imaging are lacking to date. To address this bottleneck, we developed the first laparoscopic real-time multispectral imaging (MSI) system featuring a compact and lightweight multispectral camera and the possibility to complement the conventional RGB (Red, Green, and Blue) surgical view of the patient with functional information at a video rate of 25 Hz. To account for the high inter-patient variability of human tissue, we phrase the problem of ischemia detection as an out-of-distribution (OoD) detection problem that does not rely on data from any other patient. Using an ensemble of invertible neural networks (INNs) as a core component, our algorithm computes the likelihood of ischemia based on a short (several seconds) video sequence acquired at the beginning of each surgery. A first-in-human trial performed on 10 patients undergoing partial nephrectomy demonstrates the feasibility of our approach for fully-automatic live ischemia monitoring during laparoscopic surgery. Compared to the clinical state-of-the-art approach based on indocyanine green (ICG) fluorescence, the proposed MSI-based method does not require the injection of a contrast agent and is repeatable if the wrong segment has been clamped. Spectral imaging combined with advanced deep learning-based analysis tools could thus evolve as an important tool for fast, efficient, reliable and safe functional imaging in minimally invasive surgery.