We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $\mathit{d}\ge 2$, and a connectivity phase transition whenever $\mathit{d}\ge 4$. We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.

A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.

In addition we also determine the almost sure Hausdorff dimension of the fractal set.