  We prove phase transitions for continuum percolation in a Boolean model based
on a Cox point process with nonstabilizing directing measure. The directing
measure, which can be seen as a stationary random environment for the classical
Poisson--Boolean model, is given by a planar rectangular Poisson line process.
This Manhattan grid type construction features long-range dependencies in the
environment, leading to absence of a sharp phase transition for the associated
Cox--Boolean model. The phase transitions are established under individually as
well as jointly varying parameters. Our proofs rest on discretization arguments
and a comparison to percolation on randomly stretched lattices established in
Hoffman 2005.

Probability