Dynamic metasurface antennas (DMAs) are an emerging technology for next-generation wireless base stations, distinguished by hybrid analog/digital beamforming capabilities with low hardware complexity. However, the coupling between meta-atoms is fixed in existing DMAs, which fundamentally constrains the achievable performance. Here, we introduce reconfigurable coupling mechanisms between meta-atoms, yielding finer control over the DMA's analog signal processing capabilities. This novel hardware is coined "beyond-diagonal DMA" (BD-DMA), in line with established BD-RIS terminology. We derive a physics-consistent system model revealing (correlated) "beyond-diagonal" programmability in a reduced basis. We also present an equivalent formulation in a non-reduced basis with (uncorrelated) "diagonal" programmability. Based on the diagonal representation, we propose a general and efficient mutual-coupling-aware optimization algorithm. Physics-consistent simulations validate the performance enhancement enabled by reconfigurable coupling mechanisms in BD-DMAs. The BD-DMA benefits grow with the mutual coupling strength.