Applications for the vacancy of a Domestic Partner - a cross between a housemate, critical friend and business partner - are invited.
Sex and fidelity are optional, ie neither compulsory nor prohibited. Each "transaction" is deemed to be a one-off so that both parties are kept on their toes. There would be no assumption that either party would be spending, for example, St Valentine's Day with each other as a matter of course.
So few people like this idea that the person, male or female, who says to me "My word, this is just the sort of relationship I have been looking for all my life!" without a hint of irony or sarcasm, would probably be The One.
Domestic Partnership - http://en.wikipedia.org/wiki/Domestic_partnership