Poset polytopes and pipe dreams: types C and B

The first part of this paper concerns type C. We present new explicitly defined families of algebro-combinatorial structures of three kinds: combinatorial bases in representations, Newton--Okounkov bodies of flag varieties and toric degenerations of flag varieties. All three families are parametrized by the same family of polytopes: the marked chain-order polytopes of Fang and Fourier which interpolate between the type C Gelfand--Tsetlin and FFLV polytopes. Thus, in each case the obtained structures interpolate between the well-known bases, Newton--Okounkov bodies or degenerations associated with the latter two polytopes. We then obtain similar results for type B after introducing a new family of poset polytopes to be considered in place of marked chain-order polytopes. In both types our constructions and proofs rely crucially on a combinatorial connection between poset polytopes and pipe dreams.