  Tropical geometry and the theory of Newton-Okounkov bodies are two methods
which produce toric degenerations of an irreducible complex projective variety.
Kaveh-Manon showed that the two are related. We give geometric maps between the
Newton-Okounkov bodies corresponding to two adjacent maximal-dimensional prime
cones in the tropicalization of $X$. Under a technical condition, we produce a
natural "algebraic wall-crossing" map on the underlying value semigroups (of
the corresponding valuations). In the case of the tropical Grassmannian
$Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of
a geometric map. In an Appendix by Nathan Ilten, he explains how the geometric
wall-crossing phenomenon can also be derived from the perspective of
complexity-one $T$-varieties; Ilten also explains the connection to the
"combinatorial mutations" studied by Akhtar-Coates-Galkin-Kasprzyk.
