  In this paper, we prove the existence of infinite Gibbs Delaunay Potts
tessellations for marked particle configurations. The particle systems has two
types of interaction, a so-called \emph{background potential} ensures that
small and large triangles are excluded in the Delaunay tessellation, and is
similar to the so-called hardcore potential introduced in \cite{Der08}.
Particles carry one of $q$ separate marks. Our main result is that for large
activities and high \emph{type interaction} strength the model has at least $
q$ distinct translation-invariant Gibbs Delaunay Potts tessellations. The main
technique is a coarse-graining procedure using the scales in the system
followed by comparison with site percolation on $ \Z^2 $.
