  Differential equations perturbed by multiplicative fractional Brownian
motions are considered. Depending on the value of the Hurst parameter $H$, the
resulting equation is pathwise viewed as an ODE, YDE, or RDE. In all three
regimes we show regularisation by noise phenomena by proving the strongest kind
of well-posedness with irregular drift: strong existence and path-by-path
uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift
coefficient is optimal in the sense that it agrees with the one known for the
additive case [CG16, Ger22]. In the rough regime $H\in(1/3,1/2)$ we assume
positive but arbitrarily small drift regularity for strong well-posedness,
while for distributional drift we obtain weak existence.
