In LLM-based speech translation, transcription-based chain-of-thought (CoT) suffers from a mismatch between reference transcripts used in supervised fine-tuning (SFT) and model-generated transcripts at inference. To address this, we propose joint recognition and translation fine-tuning via group relative policy optimization (GRPO). We score both transcripts and translations, with translation conditioned on model-generated transcripts, and compare three token advantage strategies. Using Qwen2.5-Omni-3B across four languages, we evaluate CoT against direct speech translation (Direct ST) under SFT and GRPO, training on CoVoST 2 and testing on CoVoST 2 and FLEURS. CoT GRPO outperforms Direct ST GRPO by 1.77 and 0.83 average BLEU points on CoVoST 2 and FLEURS. Compared to CoT SFT, GRPO boosts BLEU by 0.82 and 0.67 points and reduces word error rate (WER) by 8.8% and 7.2% relatively. These results highlight reinforcement fine-tuning as an effective method to mitigate the training-inference mismatch, jointly improving recognition and translation.

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