  Let $S$ be a compact oriented surface with boundary together with finitely
many marked points on the boundary, and let $S^\circ$ be the same surface
equipped with the opposite orientation. We consider the double $S_\mathcal{D}$
obtained by gluing the surfaces $S$ and $S^\circ$ along corresponding boundary
components. We define a notion of lamination on the double and construct
coordinates on the space of all such laminations. We show that this space of
laminations is a tropical version of the symplectic double introduced by Fock
and Goncharov. There is a canonical pairing between our laminations and the
positive real points of the symplectic double. We derive an explicit formula
for this pairing using the $F$-polynomials of Fomin and Zelevinsky.
