In this paper we study Poisson processes of so-called "fat" cylinders in hyperbolic space. As our main result we show that this model undergoes a percolation phase transition. We prove this by establishing a novel link between the fat Poisson cylinder process and semi-scale invariant random fractal models on the unit sphere and in $\mathbb{R}^d.$ As a secondary result, we show that the semi-scale invariant fractal ball model in $\mathbb{R}^d$ has a non-empty sheet phase.