The combinatorial description of the moduli space of curves in terms of ribbon graphs was one of the crucial ingredients in Kontsevich’s proof of Witten’s conjecture. The same model was used by Harer–Zagier and by Norbury to compute the Euler characteristic and the number of lattice points, respectively. In this talk, I will introduce the analogous combinatorial model for the moduli space of real curves: the moduli of metric Möbius graphs. Each such graph is equipped with a measure of non-orientability, a quantity interpolating between the complex and the real worlds. We then provide recursive formulas for the volumes and the lattice-point counts of this moduli space, refining the results of Kontsevich and Norbury. Finally, we compute the refined Euler characteristic, thereby answering a question posed by Goulden, Harer, and Jackson 25 years ago. Based on a work in progress with N. K. Chidambaram, E. Garcia-Failde, K. <a href="http://Osuga.

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