A pneumatic radial ply runflat tire (200) has a tread (210), belt structure (212), a carcass (216), and an underlay (234) disposed between the belt structure and the carcass. The carcass (216) comprises two sidewalls (226), two beads (220), and one or more radial plies (222,224). The tire (200) is characterized by the underlay (234) having reinforcement cords (236) with turns aligned parallel to the tire's equatorial plane, the cords (236) exhibiting negligible modulus of elasticity (“modulus”) below a threshold elongation and high modulus above the threshold elongation. The underlay (234) stiffens the tire during runflat operation but not during normal inflated operation.