Having shown early promise, free-space optical communications (FSO) face formidable challenges in the age of information explosion. The ever-growing demand for greater channel communication capacity is one of the challenges. The inter-channel crosstalk, which severely degrades the quality of transmitted information, creates another roadblock in the way of efficient FSO implementation. Here we advance theoretically and realize experimentally a potentially high-capacity FSO protocol that enables high-fidelity transfer of an image, or set of images through a complex environment. In our protocol, we complement random light structuring at the transmitter with a deep learning image classification platform at the receiver. Multiplexing novel, independent, mutually orthogonal degrees of freedom available to structured random light can potentially significantly boost the channel communication capacity of our protocol without introducing any deleterious crosstalk. Specifically, we show how one can multiplex the degrees of freedom associated with the source coherence radius and a spatial position of a beamlet within an array of structured random beams to greatly enhance the capacity of our communication link. The superb resilience of structured random light to environmental noise, as well as extreme efficiency of deep learning networks at classifying images guarantees high-fidelity image transfer within the framework of our protocol.