Translating as “fried potato relationship”, the term denotes an extramarital relationship which is more about one person providing (especially food) for another, rather than love. It’s comparable to the concept of a “sugar daddy” in English.
It can be considered as a type of wilde Ehe, which signifies a relationship in which the couple are not married. It has negative connotations.
However, the term is said to originate after World War II to mean simply a relationship without a marriage certificate or common residence. Often widows had such a relationship so they wouldn't have to lose their widow's pension.
Das Mädchen hat ein Bratkartoffelverhältnis mit ihm.
He is the girl’s sugar daddy.
Sie sind neulich in ein Bratkartoffelverhältnis eingegangen.
They recently started seeing each other in a way which is more materialistic than caring.