O Príncipe Herdeiro Desaparecido (2024)

Both take place during the Joseon era
Both MLs are crown prince and somehow have to be on the run outside the palace with FLs

Setting in a same era (jungjong of joseon) and about place inner politics and 2 brothers falling for the same woman

Same writers, missing crown prince is a spin-off of Bossam but there isn't any actual relation between the two dramas

Both are historical dramas in which the ML is a prince who ends up on the run outside the palace, and helped by the FL who is unaware of his identity. Both MLs have a half brother who palace people are scheming to put on the throne in his place.

Rebel: Thief who stole the people is another historical drama made by the same director of missing crown prince (Kim Jin Man)