An assembly for a turbine engine includes a turbine engine first component, a turbine engine second component and a seal assembly. The first component includes a groove and a groove surface. The second component includes a tongue that extends into the groove to a tongue surface. The seal assembly at least partially seals a gap between the groove surface and the tongue surface. The seal assembly includes a rope seal and a clip that attaches the rope seal to the tongue. The rope seal is arranged within the groove between the groove surface and the tongue surface.