Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range to each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communication networks. Our main result shows that, despite the infinite range dependencies, percolation occurs in the model when the density of the Poisson point process is greater than the critical density value of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing).