Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$.
Is there an algorithm which takes two triangulations and produces a series of flips which transforms one triangulation into another?

WHAT I KNOW:

It is known that any two triangulations are connected by flips, the proof uses moduli space and is not constructive.

The only algorithm to build this series of flips I could find is in the paper of Lee Mosher "Tiling the projective foliation space of a punctured surface" (p. 40, Corollary). The main idea is to decrease the number of intersections of two triangulations doing flips. It goes like that: one takes an arc from the first triangulation and flips the first edge of the second triangulation which intersects the arc. This procedure is supposed to decrease the number of intersections. But it doesn't work, one can easily construct a counterexample.