  Our first result realizes the toric variety of every marked chain-order
polytope (MCOP) of the Gelfand--Tsetlin poset as an explicit Gr\"obner (sagbi)
degeneration of the flag variety. This generalizes the
Sturmfels/Gonciulea--Lakshmibai/Kogan--Miller construction for the
Gelfand--Tsetlin degeneration to the MCOP setting. The key idea of our approach
is to use pipe dreams to define realizations of toric varieties in Pl\"ucker
coordinates. We then use this approach to generalize two more well-known
constructions to arbitrary MCOPs: standard monomial theories such as those
given by semistandard Young tableaux and PBW-monomial bases in irreducible
representations such as the FFLV bases. In an addendum we introduce the notion
of semi-infinite pipe dreams and use it to obtain an infinite family of poset
polytopes each providing a toric degeneration of the semi-infinite
Grassmannian.
