An apparatus is provided for a gas turbine engine. This engine apparatus includes a shroud, and the shroud includes a wall and a bleed passage. The wall includes an interior surface and an exterior surface. The wall extends circumferentially about an axis. The wall extends depthwise between the interior surface and the exterior surface. The interior surface forms a peripheral boundary of a flowpath that extends along the shroud. The bleed passage includes an inlet orifice and an outlet orifice. The bleed passage extends through the shroud between the inlet orifice and the outlet orifice. The inlet orifice is disposed in the interior surface and fluidly couples the flowpath to the bleed passage. At least a downstream section of the bleed passage circumferentially tapers as the downstream section of the bleed passage extends within the wall towards the outlet orifice.