In this paper, the sensitivity of Atmospheric and Oceanic Coupled Models (AOCMs) to the associated coupling method is investigated. We propose the adaptation of a Schwarz- like domain decomposition method to AOCMs. We show that this kind of iterative process ensures a good consistency of the coupled solution across the air-sea interface, while usual ad-hoc algorithmic approaches do not properly satisfy this requirement. Indeed, those ad- hoc methods correspond to only one iteration of a Schwarz-like iterative method, without reaching a converged state. Until now, it was assumed that this lack of consistency does not affect significantly the physical properties of the solution. The relevancy of this statement is assessed in the context of a coupling between a mesoscale atmospheric model (WRF) and a regional oceanic model (ROMS) in a realistic configuration aiming at simulating the genesis and propagation of tropical cyclone Erica. Sensitivity tests to the coupling method are carried out in an ensemblist approach. We show that with a mathematically consistent coupling the spread of the ensemble is reduced thus suggesting that there is room for further improvements in the formulation of AOCMs at a mathematical and numerical level.