A shroud assembly for a gas turbine engine includes a plurality of shroud segments that are attached to a shroud support with an inter-segment joint defined between shroud segments. The shroud assembly also includes a cooling flow path cooperatively defined by the shroud support and the first shroud segment. The cooling flow path includes an internal cooling passage within the shroud segments. The cooling flow path includes an outlet chamber configured to receive flow from the internal cooling passage. The shroud assembly additionally includes a seal arrangement that extends across the inter-segment joint. The seal arrangement, the first shroud segment, and the second shroud segment cooperatively define a seal chamber that is enclosed.