Diagnosis
The filters are likely saturated with contaminants and need cleaning, or the system itself (chambers, spigot) needs a thorough sanitizing. Bacteria or biofilm can grow in the standing water, affecting taste.
How to Fix It
First, scrub your filters as described for slow flow. Then, clean the chambers. Use a mild bleach solution (1 teaspoon household bleach per gallon of water), fill the chambers, let sit for a few minutes, then rinse EXTENSIVELY. Clean the spigot internals too. Re-prime the cleaned filters before reinstalling.
If That Didn't Work
If the off-taste persists, your source water might have contaminants the Black Berkey filters don't target as effectively, like certain minerals or salts. In rare cases, a filter can develop an internal issue, but taste issues are usually solved by cleaning.
Heads Up
Storing filtered water in the lower chamber for more than a day or two can lead to stale taste. It's a filter, not a refrigerator. For best taste, drink the water within a day of filtering.