We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter $\mathcal{H}>\frac{1}{3}$. The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of $\mathcal{H}$. We prove convergence in law of the slow component to a Navier–Stokes system with an additional It\^o-Stokes drift when $\mathcal{H}<\frac{1}{2}$. In contrast, for $\mathcal{H}\in (\frac{1}{2},1)$, the limit equation features only a transport noise driven by a rough path.