Audio and omni-modal large language models exhibit impressive cross-modal reasoning capabilities. However, applying standard reinforcement learning post-training algorithms to these models exposes a critical structural vulnerability: methods like GRPO apply uniform policy gradients across all tokens, ignoring their unequal dependence on the non-text source modality. This exacerbates late-stage modality collapse during extended chain-of-thought generation, where models progressively abandon the primary source signal in favor of compressed textual priors, leading to confident but ungrounded hallucinations. To address this, we introduce Modality-Aware Policy Optimization (MAPO), a novel dual-branch reinforcement learning framework. First, MAPO dynamically concentrates the policy gradient on modality-critical tokens using a modality relevance mask, which is derived from the cross-modal differential entropy between an audio-ablated reference and the multimodal policy. Second, it integrates an auxiliary attention loss branch that applies a targeted, temporally scaled penalty to the model's internal attention distributions. This ensures the model actively sustains cross-modal grounding deep into the reasoning trace. Evaluations on complex audio reasoning benchmarks demonstrate that MAPO substantially improves long-horizon reasoning fidelity and multimodal instruction following, achieving highly competitive performance and setting new state-of-the-art results on several key benchmarks among open-weight models. By relying strictly on native statistical signals rather than domain-specific inductive biases, MAPO offers a promising foundation for mitigating epistemic collapse across diverse multimodal systems.