As the complexity of our energy systems increases, cross-domain supervisory controller development gets increasingly difficult. In this paper, co-simulation is used to analyze supervisory control strategies for a multi-physics green hydrogen offshore wind turbine (GHOWT), including a multi-stack electrolyzer (mELY). For the cross-domain model composition, functional mock-up units (FMUs) are exported from various simulation platforms and executed in Python. We explore the operational characteristics of four rule-based energy management system (EMS) strategies while considering auxiliary component dynamics. The dynamics are important since in the off-grid application, fresh water needs to be produced by reverse osmosis and balance of stack units (e.g. pumps and heating) can only be operated by generated wind power. A case study with a 15MW wind turbine and three single electrolyzers (sELYs) is carried out over one week of turbulent power generation. Fresh water supply is only found to be limiting in the start-up phase with the chosen system configuration. The Equal distribution method with degradation feedback (EQx) is favorable considering all key performance indicators (KPIs) which are hydrogen production, energy use and degradation. Nevertheless, in all cases, excessive on-idle switching events cause uneconomic electrolyzer degradation, which is calling for optimized control approaches like model predictive control.