In the southern hemisphere you would undershoot when turning towards a southerly heading and overshoot towards a northerly heading, and the opposite in the northern hemisphere. Why is this?
There is a good video that goes over the topic here ~2:45 into the video. It has to do with the way a magnetic compass responds to turns. When turning north the compass lags when compared to the tun of the physical aircraft so you need to overshoot the heading based on what the compass is showing. When turning south the compass leads the turn and you must roll out early to make your desired heading.
This site offers a nice explanation of why this happens in the compas
Northerly turning error is due to the mounting of the compass. Since the card is balanced in fluid, when the aircraft turns, the card is also banked as a result of centrifugal force. While the card is banked, the vertical component of the Earth's magnetic field causes the north-seeking ends of the compass to dip to the low side of the turn. When making a turn from a northerly heading, the compass briefly gives an indication of a turn in the opposite direction. When making a turn from the south, it gives an indication of a turn in the correct direction but at a faster rate.
Similar explanations show up in this thread. Simply put its not the turn that causes the error but the banking of the aircraft. By rolling the compass housing to one side we allow the compass to tilt differently than it would in level flight. Since the pole is generally beneath the aircraft the roll allows the compass to pitch in a slightly different way causing the error we see.