  For any cluster algebra whose underlying combinatorial data can be encoded by
a bordered surface with marked points, we construct a geometric realization in
terms of suitable decorated Teichmueller space of the surface. On the geometric
side, this requires opening the surface at each interior marked point into an
additional geodesic boundary component. On the algebraic side, it relies on the
notion of a non-normalized cluster algebra and the machinery of tropical lambda
lengths.
  Our model allows for an arbitrary choice of coefficients which translates
into a choice of a family of integral laminations on the surface. It provides
an intrinsic interpretation of cluster variables as renormalized lambda lengths
of arcs on the surface. Exchange relations are written in terms of the shear
coordinates of the laminations, and are interpreted as generalized Ptolemy
relations for lambda lengths.
  This approach gives alternative proofs for the main structural results from
our previous paper, removing unnecessary assumptions on the surface.
