  We consider a dependent percolation model on the square lattice
$\mathbb{Z}^2$. The range of dependence is infinite in vertical and horizontal
directions. In this context, we prove the existence of a phase transition. The
proof exploits a multi-scale renormalization argument that is defined once the
environment configuration is suitably good and, which, together with the main
estimate for the induction step, comes from Kesten, Sidoravicius and Vares (To
appear in {\em Electronic Journal of Probability}, (2022)). This work was
inspired by de Lima (Ph.D.Thesis, \emph{Informes de Matem\'atica. IMPA},
S\'erie C-26/2004) where the simpler case of a deterministic environment was
considered. It has various applications, including an alternative proof for the
phase transition on the two dimensional random stretched lattice proved by
Hoffman ({\em Comm. Math. Phys.} {\bf 254}, 1-22 (2005)).
