  We construct a new (cyclic) operad of `mosaics' defined by polygons with
marked diagonals. Its underlying (aspherical) spaces are the sets of real
points of the moduli space of punctured Riemann spheres, which are naturally
tiled by Stasheff associahedra. We (combinatorially) describe them as iterated
blow-ups and show that their fundamental groups form an operad with
similarities to the operad of braid groups.
