Simultaneous speech translation (SimulST) aims to translate ongoing speech with low latency while preserving translation quality. Recent studies have adopted large language models (LLMs) for SimulST and formulated it as a multi-turn dialogue task for efficient generation, achieving state-of-the-art performance. However, existing dialogue-style SimulST approaches typically rely on specialized training schemes to acquire simultaneous translation behavior, either using synthesized read/write trajectories, which may provide suboptimal training signals, or jointly optimizing translation quality and latency, whose competing objectives may cause optimization instability. In this paper, we introduce SimPOTrans, a reinforcement learning (RL) framework that enables offline speech translation models to self-adapt to dialogue-style SimulST. Through a simple two-stage RL post-training pipeline, SimPOTrans rewards an offline model for exhibiting translation behaviors beneficial to simultaneous scenarios when given partial speech inputs, thereby enabling offline-to-simultaneous self-adaptation without synthesized read/write trajectories or explicit latency optimization. At inference time, SimPOTrans requires no external decision policy and enables flexible quality--latency control by simply adjusting the chunk size of speech input. Experiments on the ACL 60/60 and MCIF benchmarks across three language directions (En$\rightarrow$De, En$\rightarrow$Zh, and En$\rightarrow$Ja) demonstrate that SimPOTrans outperforms several strong baselines, achieving a superior quality--latency trade-off.

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