  We find new examples of the $P=W$ identity of de Cataldo-Hausel-Migliorini by
studying cluster varieties. We prove that the weight filtration of 2D cluster
varieties correspond to the perverse filtration of elliptic fibrations which
are deformation equivalent to the elliptic fibrations of types $I_b$ or $2I_b$.
These examples do not arise from character varieties or moduli of Higgs
bundles.
