  A Gr\"obner basis computation for the Weyl algebra with respect to a tropical
term order and by using a homogenization-dehomogenization technique is
sufficiently sluggish. A significant number of reductions to zero occur. To
improve the computation, a tropical F5 algorithm is developed for this context.
As a member of the family of signature-based algorithms, this algorithm keeps
track of where Weyl algebra elements come from to anticipate reductions to
zero. The total order for ordering module monomials or signatures in this paper
is designed as close as possible to the definition of the tropical term order.
As in Vaccon et al. (2021), this total order is not compatible with the
tropical term order.
