Tropical bases by regular projections

We consider the tropical variety $\mathcal{T}(I)$ of a prime ideal $I$ generated by the polynomials $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_r$ and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that $I$ has a short tropical basis of cardinality at most $r+\text{codim}I+1$ at the price of increased degrees, and we provide a computational description of these bases.