  We establish a sprinkled decoupling inequality for increasing events of
Gaussian vectors with an error that depends only on the maximum pairwise
correlation. As an application we prove the non-triviality of the percolation
phase transition for Gaussian fields on $\mathbb{Z}^d$ or $\mathbb{R}^d$ with
(i) uniformly bounded local suprema, and (ii) correlations which decay at least
polylogarithmically in the distance with exponent $\gamma > 3$; this expands
the scope of existing results on non-triviality of the phase transition,
covering new examples such as non-stationary fields and monochromatic random
waves.
