Click here to download a tarball of Bertini input files and a shell script for running the algorithm in the paper.
In this case, we know the location of the ramification points, so we can take the liberty of hard-coding reasonable radii for the various monodromy loops.
Ultimately, this is a rather simple example, but it at least provides the Bertini input files for the example in the article.
These computations make use of Bertini. Please visit here to download the latest version of Bertini.