A shroud assembly for a gas turbine engine comprising: a shroud support that extends arcuately about an axis; a plurality of shroud segments that are attached to the shroud support and that are arranged annularly about the axis at different circumferential positions with respect to the axis, the plurality of shroud segments including a first shroud segment and a second shroud segment, an inter-segment joint defined circumferentially between the first and second shroud segments, the inter-segment joint including a leading edge and a trailing edge that are separated apart along an axial direction; a cooling flow path cooperatively defined by the shroud support and the first shroud segment, the cooling flow path including an internal cooling passage defined within the first shroud segment, the cooling flow path including an outlet chamber configured to receive flow from the internal cooling passage, the outlet chamber defined between the first shroud segment and the shroud support and fluidly connected to the inter-segment joint; and characterized in that a seal arrangement that extends across the inter-segment joint, the seal arrangement including a first sealing member and a second sealing member; wherein, at a first portion of the seal arrangement, the first sealing member and the second sealing member overlap and are in abutting contact; wherein, at a second portion of the seal arrangement, the first sealing member and the second sealing member are separated apart at a distance; wherein, at a third portion of the seal arrangement, the first sealing member and the second sealing member overlap and are in abutting contact; wherein the second portion is disposed between the first portion and the third portion and defines a seal chamber that is enclosed, the seal chamber disposed proximate the trailing edge and spaced apart at an axial distance from the leading edge. a shroud support that extends arcuately about an axis; a plurality of shroud segments that are attached to the shroud support and that are arranged annularly about the axis at different circumferential positions with respect to the axis, the plurality of shroud segments including a first shroud segment and a second shroud segment, an inter-segment joint defined circumferentially between the first and second shroud segments, the inter-segment joint including a leading edge and a trailing edge that are separated apart along an axial direction; a cooling flow path cooperatively defined by the shroud support and the first shroud segment, the cooling flow path including an internal cooling passage defined within the first shroud segment, the cooling flow path including an outlet chamber configured to receive flow from the internal cooling passage, the outlet chamber defined between the first shroud segment and the shroud support and fluidly connected to the inter-segment joint; and characterized in that a seal arrangement that extends across the inter-segment joint, the seal arrangement including a first sealing member and a second sealing member; wherein, at a first portion of the seal arrangement, the first sealing member and the second sealing member overlap and are in abutting contact; wherein, at a second portion of the seal arrangement, the first sealing member and the second sealing member are separated apart at a distance; wherein, at a third portion of the seal arrangement, the first sealing member and the second sealing member overlap and are in abutting contact; wherein the second portion is disposed between the first portion and the third portion and defines a seal chamber that is enclosed, the seal chamber disposed proximate the trailing edge and spaced apart at an axial distance from the leading edge.