Currently, it is possible to generically solve for all Belyi maps (using homotopy continuation) for a given genus 0 dessin d'enfant passport and normalisation. Sometimes it can do higher genera. This version (a pre-release to a versin to come around May 2023) makes stock usage of the brilliant HomotopyContinuation.jl from Sascha Timme and Paul Breiding. With the expected publication of my thesis will come a version maing specific optimisation to solving Belyi maps --- making particular use of intrinsic properties of the systems that arise as a result (in particular targetting the mixed volumne calculations. Should little of this jargon make sense to you then idk go read a thesis (there are literally hundereds) it's very interesting stuff.
In case you are attempting to solve for higher genus maps, you will often end up with non-square systems, thus necessitating the use of parametric homotopy continuation, which can be done via monodromy methods (although the inclusion of this into the package as given is yet to come). A currently unpublished (but coming soon to a GitHub near you) Python package handles the exactification of these rational approximations into algebraic numbers, the fields they define, the plotting of the dessins, and most importantly, the determination of Galois conjugacy under the action of the absolute Galois group of the rationals. First, however, housecleaning is in order before it should see the light of day.