  To each tagged triangulation of a surface with marked points and non-empty
boundary we associate a quiver with potential, in such a way that whenever we
apply a flip to a tagged triangulation, the Jacobian algebra of the QP
associated to the resulting tagged triangulation is isomorphic to the Jacobian
algebra of the QP obtained by mutating the QP of the original one. Furthermore,
we show that any two tagged triangulations are related by a sequence of flips
compatible with QP-mutation. We also prove that for each of the QPs
constructed, the ideal of the non-completed path algebra generated by the
cyclic derivatives is admissible and the corresponding quotient is isomorphic
to the Jacobian algebra. These results, which generalize some of the second
author's previous work for ideal triangulations, are then applied to prove
properties of cluster monomials, like linear independence, in the cluster
algebra associated to the given surface by Fomin-Shapiro-Thurston (with an
arbitrary system of coefficients).
