This is my current, and assuredly incomplete list of fertile potato varieties, or those that will frequently produce berries. This is separated into continents where they are or have been grown. Feel free to add to this list. If you are looking to grow potatoes that produce their own berries, locating a seed tuber source for some of those on this list would be a good starting point.
North America (USA and/or CANADA)
ALL BLUE / CONGO
ATLANTIC
BORA VALLEY
BOULDER / PI 656384
CARIBE
CHIEFTAIN
DESIREE
GERMAN BUTTERBALL
HUCKLEBERRY GOLD
KATAHDIN
KING HARRY
LA RATTE
MARIS PIPER
PURPLE MAJESTY
RED GOLD
YUKON GOLD
Europe (EU)
ATLANTIC
BLUE BELLE
BOXER
CARA
CHIEFTAIN
CLAUSTAR
COMLE (1964)
CONGO / ALL BLUE
DESIREE
DR MCINTOSH
GRANOLA
INCA BELLE
INCA SUN
HIGHLAND BURGUNDY RED
HINDENBURG
KAMERAZ
KELTIA
KERPONDY
LA RATTE
LEKSANDS VIT
MARIS BARD
MARIS PIPER
MARABELLE
MARFONA
MAYAN GOLD
MIRAKEL
MORENE
NEMES ROZSA
ORION (1947)
PENTLAND DELL
PURPLE MAJESTY
RECORD
REVELINO
ROOSTER
SALAD BLUE
SATURNA
STINA
TONE
VALOR
VESTAR
South America
CHAPOSA (Bolivia)
FRIPAPA (Ecuador)
PERRICHOLI / KABALE (Bolivia, Burundi, Uganda, Vietnam, Ethiopia, Guatemala y Peru)
Africa
MURUTA (Burundi)
PERRICHOLI / KABALE (Bolivia, Burundi, Uganda, Vietnam, Ethiopia, Guatemala y Peru)
ENFULA ( ?)
ABADAMU - (Ethiopia)
Asia
GRANOLA
IGOROTA / PO3
PERRICHOLI / KABALE (Bolivia, Burundi, Uganda, Vietnam, Ethiopia, Guatemala y Peru)
