  Marked chain-order polytopes are convex polytopes constructed from a marked
poset, which give a discrete family relating a marked order polytope with a
marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of
type A, and realize the associated marked chain-order polytopes as
Newton-Okounkov bodies of the flag variety. Our realization connects previous
realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg
polytopes as Newton-Okounkov bodies in a uniform way. As an application, we
prove that the flag variety degenerates into the irreducible normal projective
toric variety corresponding to a marked chain-order polytope. We also construct
a specific basis of an irreducible highest weight representation which is
naturally parametrized by the set of lattice points in a marked chain-order
polytope.
