  We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective
variety over a discretely or trivially valued field. First, we give an example
of a Green's function, associated to a divisorial valuation, which is not Q-PL
(i.e. not a model function in the discretely valued case). Second, we produce
an example of a function whose Monge-Amp\`ere measure is a finite atomic
measure supported in a dual complex, but which is not invariant under the
retraction associated to any snc model. This answers a question by Burgos Gil
et al in the negative. Our examples are based on geometric constructions by
Cutkosky and Lesieutre, and arise via base change from Green's functions over a
trivially valued field; this theory allows us to efficiently encode the Zariski
decomposition of a pseudoeffective numerical class.
