  A universal analytic Gr{\"o}bner basis (UAGB) of an ideal of a Tate algebra
is a set containing a local Gr{\"o}bner basis for all suitable convergence
radii. In a previous article, the authors proved the existence of finite UAGB's
for polynomial ideals, leaving open the question of how to compute them. In
this paper, we provide an algorithm computing a UAGB for a given polynomial
ideal, by traversing the Gr{\"o}bner fan of the ideal. As an application, it
offers a new point of view on algorithms for computing tropical varieties of
homogeneous polynomial ideals, which typically rely on lifting the computations
to an algebra of power series. Motivated by effective computations in tropical
analytic geometry, we also examine local bases for more general convergence
conditions, constraining the radii to a convex polyhedron. In this setting, we
provide an algorithm to compute local Gr{\"o}bner bases and discuss obstacles
towards proving the existence of finite UAGBs. CCS CONCEPTS $\bullet$ Computing
methodologies $\rightarrow$ Algebraic algorithms.
