Machine translation has long been a central task in natural language processing. With the rapid advancement of large language models (LLMs), there has been remarkable progress in translation quality. However, fully realizing the translation potential of LLMs remains an open challenge. Recent studies have explored multi-agent systems to decompose complex translation tasks into collaborative subtasks, showing initial promise in enhancing translation quality through agent cooperation and specialization. Nevertheless, existing multi-agent translation frameworks largely neglect foundational insights from cognitive translation studies. These insights emphasize how human translators employ different cognitive strategies, such as balancing literal and free translation, refining expressions based on context, and iteratively evaluating outputs. To address this limitation, we propose a cognitively informed multi-agent framework called TACTIC, which stands for T ranslation A gents with Cognitive- T heoretic Interactive Collaboration. The framework comprises six functionally distinct agents that mirror key cognitive processes observed in human translation behavior. These include agents for drafting, refinement, evaluation, scoring, context reasoning, and external knowledge gathering. By simulating an interactive and theory-grounded translation workflow, TACTIC effectively leverages the full capacity of LLMs for high-quality translation. Experimental results on diverse language pairs from the FLORES-200 and WMT24 benchmarks show that our method consistently achieves state-of-the-art performance. Using DeepSeek-V3 as the base model, TACTIC surpasses GPT-4.1 by an average of +0.6 XCOMET and +1.18 COMETKIWI-23. Compared to DeepSeek-R1, it further improves by +0.84 XCOMET and +2.99 COMETKIWI-23. Code is available at https://github.com/weiyali126/TACTIC.