  We introduce Eulerian maps with blocked edges as a general way to implement
statistical matter models on random maps by a modification of intrinsic
distances. We show how to code these dressed maps by means of mobiles, i.e.
decorated trees with labeled vertices, leading to a closed system of recursion
relations for their generating functions. We discuss particular solvable cases
in detail, as well as various applications of our method to several statistical
systems such as spanning trees on quadrangulations, mutually excluding
particles on Eulerian triangulations or the Ising model on quadrangulations.
