  We investigate the Voronoi diagrams with respect to an asymmetric tropical
distance function, also for infinite point sets. These turn out to be much
better behaved than the tropical Voronoi diagrams arising from the standard
tropical distance, which is symmetric. In particular, we show that the
asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power
diagrams over fields of real Puiseux series. Our results are then applied to
rational lattices and Laurent monomial modules.
