Text Image Machine Translation (TIMT)—the task of translating textual content embedded in images—is critical for applications in accessibility, cross-lingual information access, and real-world document understanding. However, TIMT remains a complex challenge due to the need for accurate optical character recognition (OCR), robust visual-text reasoning, and high-quality translation, often requiring cascading multi-stage pipelines. Recent advances in large-scale Reinforcement Learning (RL) have improved reasoning in Large Language Models (LLMs) and Multimodal LLMs (MLLMs), but their application to end-to-end TIMT is still underexplored. To bridge this gap, we introduce MT³, a novel Multi-Task RL framework to specialize MLLMs into end-to-end expert TIMT models. MT³ adopts a synergistic multi-task optimization paradigm targeting three key sub-skills: text recognition, context-aware reasoning, and translation. It is trained using a novel multi-mixed reward mechanism that provides fine-grained feedback, fostering a controllable and transparent optimization process. Furthermore, to facilitate the evaluation of TIMT in authentic cross-cultural and real-world social media contexts, we introduced XHSPost, the first social media TIMT benchmark. Our MT³-7B-Zero achieves state-of-the-art results on the latest in-domain MIT-10M benchmark, outperforming strong baselines such as Qwen2.5-VL-72B and InternVL2.5-78B by notable margins across multiple metrics. Additionally, the model shows strong generalization to out-of-distribution language pairs and datasets. In-depth analyses reveal how multi-task synergy, reinforcement learning initialization, curriculum design, and reward formulation contribute to advancing MLLM-driven TIMT.