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Multispectral optoacoustic tomography (MSOT) is a high-resolution functional imaging modality that can non-invasively access a broad range of pathophysiological phenomena by quantifying the contrast of endogenous chromophores in tissue. Real-time imaging is imperative to translate MSOT into clinical imaging, visualize dynamic pathophysiological changes associated with disease progression, and enable in situ diagnoses. Model-based reconstruction affords state-of-the-art optoacoustic images; however, the image quality provided by model-based reconstruction remains inaccessible during real-time imaging because the algorithm is iterative and computationally demanding. Deep learning affords faster reconstruction, but the lack of ground truth training data can lead to reduced image quality for in vivo data. We introduce a framework, termed DeepMB, that achieves accurate optoacoustic image reconstruction for arbitrary input data in 31 ms per image by expressing model-based reconstruction with a deep neural network. DeepMB facilitates accurate generalization to experimental test data through training on signals synthesized from real-world images and ground truth images generated by model-based reconstruction. The framework affords in-focus images for a broad range of anatomical locations because it supports dynamic adjustment of the reconstruction speed of sound during imaging. Furthermore, DeepMB is compatible with the data rates and image sizes of modern multispectral optoacoustic tomography scanners. We evaluate DeepMB on a diverse dataset of in vivo images and demonstrate that the framework reconstructs images 1000 times faster than the iterative model-based reference method while affording near-identical image qualities. Accurate and real-time image reconstructions with DeepMB can enable full access to the high-resolution and multispectral contrast of handheld optoacoustic tomography.