I have a similar situation with unsugared black tea

You should brew tea properly. The bitterness in black tea comes from tannins and you can easily control their extraction during brewing.

For black tea:

• Use good quality loose-leaf tea, preferably the orange pekoe cut. The tea dust in teabags oversteeps (and becomes bitter) very quickly.

• Time the steeping of the tea leaves in boiling water. The usually recommended five minutes is often too long (if you'll be drinking your tea without milk). Try three and a half minutes to start, then adjust as desired. The less time you steep, the less bitter will the tea be (while caffeine generally extracts in the first 30 seconds or so).

• Once the tea has steeped for the proper time, pour the tea off the leaves into another teapot (or a cup, etc.). Do not let tea sit on the leaves.

• Experiment with different teas as well. Tea from Assam (often sold as Irish Breakfast) is the most tannic. Tea from Darjeeling, for example, is much less so.