  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 $d\geq 2,$ and
a connectivity phase transition whenever $d\geq 4.$ 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.
