Purpose <p>The emerging concept of odd elasticity, characterized by asymmetric stress tensors, introduces atypical mechanical responses that challenge conventional symmetries. While linear non-Hermitian systems have been extensively studied, the nonlinear dynamics within non-Hermitian metamaterials remain largely unexplored.</p> Methods <p>This work bridges this gap by designing and theoretically analyzing an active, weakly nonlinear acoustic metamaterial. Our proposed microstructure, a mass-in-mass chain with a programmable feed-forward control system, introduces active forces that generate unbalanced off-diagonal stiffness terms, leading to a non-Hermitian system that exhibits odd elasticity.</p> Results <p>Our analysis reveals remarkable non-Hermitian wave phenomena, including nonreciprocal transmission and mode-dependent unidirectional amplification. Crucially, we uncover a novel anomalous wave back-propagation phenomenon arising from the synergistic interplay of nonlinearity and odd elasticity, where an amplified wave in one mode triggers a subharmonic that propagates in the reverse direction.</p> Conclusion <p>Our analysis reveals remarkable non-Hermitian wave phenomena, including nonreciprocal transmission and mode-dependent unidirectional amplification. Crucially, we uncover a novel anomalous wave back-propagation phenomenon arising from the synergistic interplay of nonlinearity and odd elasticity, where an amplified wave in one mode triggers a subharmonic that propagates in the reverse direction.</p>