Tower B is a 19th century reconstruction of a tower that was previously demolished. We know that there was an original tower in this position, as it is shown on maps from as late as 1682. It does not appear on maps dated 1700, so must have been demolished sometime after the late 1680's.
Above. Tower B looking North up Marygate.
Tower B lies about two hundred feet North of Tower A, is slightly taller than its sibling and projects an identical five feet from the face of the wall.
If the 19th century reconstruction of the tower is correct, and Tower B was the same as Tower A, the original would have been built around 1266. It is not known why the original was demolished, or even if it collapsed due to neglect or siege damage.