As said, proper fit, and multiple hand positions are the key.
Because you tend to move around less on a tandem it becomes a bigger issue. Try to keep your upper body relaxed, keeping your elbows bent, and light pressure on your hands.
Watch how your wrist is cocked. If your wrist is bent back it can entrap the nerve, so try to keep a relaxed angle at your wrist.
also pressure right at the base of your palms can entrap the nerve. So try moving around what portion of your palm presses on the bar.
One position that relieves pressure on the nerves is to put your fingers over the brake hoods, with the hood resting between your index and middle finger. While it's not very comfratable, it gives your palms a complete break for a minute or two.
Ride one handed gently shaking the other hand for a minute or so.
Make sure the bike fits, relax, and move your hands around a lot, and you should be fine