Recent research has proposed media-based modulation (MBM) as a method to reduce the hardware complexity of wireless communications systems and therefore also achieve a reduction of the associated cost. In this work, we propose an MBM system based on a novel asymmetric signal constellation consisting of scaled and shifted Eisenstein integers. The constellation is generated by phase shifts induced by a reconfigurable intelligent antenna, where the magnitudes are modulated by turning on or off certain numbers of reflecting elements. At the receiver, a uniform linear antenna array is used to capture the incident electromagnetic planar wave. Robust estimation techniques, such as the median, the Weiszfeld algorithm, and the Sq-estimator are employed to recover the constellation points. A novel gain control scheme is proposed together with a phase offset detection method based on circular cross-correlation. Furthermore, complex-valued convolutional neural networks are used to decode the received signal. We also consider the performance of our system under impulse noise caused by voltage transients in addition to additive white Gaussian noise and show superior performance vis-a-vie a generic 64-QAM modulation scheme and a brute-force arithmetic method based on the four-quadrant arctan function and the median.