  It is already known that order polytopes and chain polytopes are always
2-level polytopes. In general, this is not true for marked order and marked
chain polytopes. We study the geometry of marked order polytopes, marked chain
polytopes, and marked chain-order polytopes, providing a comprehensive
characterisation of 2-levelness for these polytopes. Furthermore, we present an
exact formula for the Ehrhart polynomial of marked order polytopes. Because of
their connection to marked chain and marked chain-order polytopes, this
polynomial is also the Ehrhart polynomial of these polytopes.
