  An embedded variety is said to be well-poised when the associated initial
ideal degenerations coming from points of the tropical variety are reduced and
irreducible. Varieties with a well-poised embedding admit a large collection of
explicitly constructible Newton-Okounkov bodies.
  This paper aims to study the well-poised property under torus quotients. Our
first result states that GIT quotients of normal well-poised varieties by
quasi-tori also have well-poised embeddings. As an application, we show that
several Hassett spaces, $\overline{M}_{0,\beta}$, are well-poised under
Alexeev's embedding. Conversely, given an affine $T$-variety $X$ with
polyhedral divisor $\mathfrak{D}$ on a well-poised base $Y$, we construct an
embedding of $X \subseteq \mathbb{A}^N$ and provide conditions on $Y$ and
$\mathfrak{D}$ which if met, imply $X$ is well-poised under this embedding.
Then we show that any affine arrangement variety meets the specified criteria,
generalizing results of Ilten and the second author for rational complexity 1
varieties. Using this result, we explicitly compute many Newton-Okounkov cones
of $X$ and provide a criterion for the associated toric degenerations to be
normal. Our final application combines these two results to show that
hypertoric varieties have well-poised embeddings.
