  In recent times, a wide variety of combinatorics has been introduced in order
to solve problems from algebraic geometry. Newton-Okounkov bodies and tropical
geometry are two such combinatorial theories. As shown by Kaveh and Manon,
there is a certain correspondence between these two. Building on this
correspondence, and exploiting the link of both theories to toric
degenerations, Harada and Escobar obtained their wall-crossing result for prime
cones. This result states that moving between two adjacent prime maximal cones
in a tropical variety corresponds to a mutation between the associated
Newton-Okounkov bodies of these cones. In this thesis, we provide a method for
applying the wall-crossing result to non-prime cones. Our approach uses a
procedure developed by Bossinger, Lamboglia, Mincheva and Mohammadi in order to
compute an embedding which changes a tropical variety in such a way that a
non-prime cone becomes prime. Assuming that adjacent cones stay adjacent, the
wall-crossing result can then be applied in this new embedding. Building on
computations by Clarke, Mohammadi and Zaffalon, we show that for Gr(3,6), this
approach works. We compute the new embedding and its tropicalization in this
case, and study the relation between cones in the new embedding and the
original embedding.
