We propose a recurrent neural-network for real-time reconstruction of acoustic camera spherical maps. The network, dubbed DeepWave, is both physically and algorithmically motivated: its recurrent architecture mimics iterative solvers from convex optimisation, and its parsimonious parametrisation is based on the natural structure of acoustic imaging problems. Each network layer applies successive filtering, biasing and activation steps to its input, which can be interpreted as generalised deblurring and sparsification steps. To comply with the irregular geometry of spherical maps, filtering operations are implemented efficiently by means of graph signal processing techniques. Unlike commonly-used imaging network architectures, DeepWave is moreover capable of directly processing the complex-valued raw microphone correlations, learning how to optimally back-project these into a spherical map. We propose moreover a smart physically-inspired initialisation scheme that attains much faster training and higher performance than random initialisation. Our real-data experiments show DeepWave has similar computational speed to the state-of-the-art delay-and-sum imager with vastly superior resolution. While developed primarily for acoustic cameras, DeepWave could easily be adapted to neighbouring signal processing fields, such as radio astronomy, radar and sonar.