The global maritime industry is facing the goal to achieve "carbon neutrality", and substituting traditional ship fuel with clean energy is the most important solution. Whether to promote clean-energy ships through state governance or unified international supervision to achieve carbon neutrality has been a long-standing debate. This study employs both a tripartite evolutionary game model and a quadrilateral evolutionary game model to compare the effectiveness of state governance and unified international supervision in promoting clean-energy ships. The tripartite model simulates interactions between local governments, shipping companies, and shippers, revealing that high subsidy costs reduce local governments' incentives to promote clean-energy ships. The quadrilateral model introduces international supervision, demonstrating that a unified international oversight mechanism can accelerate the global maritime industry's progress toward carbon neutrality. Numerical simulations indicate that lowering the purchase costs of clean-energy ships and introducing government subsidies significantly influence the adoption of clean-energy ships, while supervision costs impact the effectiveness of international regulations. These findings provide critical insights for policymakers aiming to balance financial incentives and global supervision strategies to foster low-carbon development in the maritime sector.