There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under homeomorphism are in one-to-one correspondence with equivalence classes of simplicial complexes under Pachner moves (not mentioning technical details, e.g. that we should add "piece-wise linear" to both manifolds and complexes).
A very similar construction can be made for spin manifolds and spin homeomorphisms: In e.g. https://arxiv.org/abs/1505.05856 they give a discrete representant of the second Stiefel-Whitney class for $3$ dimensional simplicial complexes as a $1$-cycle in the complex. Then a discrete representation of the spin structure is given by a $2$-cycle that has this $1$-cycle as boundary.
My question: Is there any construction similar in spirit for manifolds and conformal maps (in $2$ dimensions)? I.e. is there any kind of $2$ dimensional combinatorial structure and a set of local moves such that equivalence classes of the combinatorial structure under local moves correspond to equivalence classes of manifolds under conformal maps?
One could for example still use simplicial complexes and restrict the set of moves such that a combinatorial analogue of "angles are preserved" holds, or add a new kind of structure to the combinatorial description as in the case of spin structures above.