What is it called when you're living with your boyfriend/girlfriend but you are not married yet?
What is it called when a couple decides they don't want to get married but plays the role of a married couple?
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You would say such couples were cohabiting.
cohabit: intransitive verb To live together as or as if a married couple - Merriam-Webster
They cohabited in a small apartment in the city.
It all depends upon the perceived state of your relationship and how much you want to reveal.
A simple "Boyfriend/girlfriend" (sometimes even just "friend") is acceptable even for a couple living together.
In some places, the law gives rights to couples who live together more than a certain amount of time, also called common-law marriage or a de facto relationship. Members of such couples refer to each other as "common-law husband/wife" or even just "husband/wife", even if they're not actually married.
And finally, people who have gone through the engagement ritual will call each other fiancé/fiancée even if "setting a date" for the wedding is a far-off proposition.
Among my circle of friends in California, we (somewhat tongue-in-cheek) call it an MLR (marriage-like relationship) when two people are essentially living a married lifestyle but have not taken formal marriage vows. This is a term we (as far as I know) made up and that is not used anywhere else, but if you like it, feel free to spread it around.
Shacking up is an informal way of saying cohabiting. On a form you might fill in cohabiting. Talking with friends or family you might say "shacking up".
Shack up: Verb: shack up
Share living quarters; usually said of people who are not married and live together as a couple - cohabit, live together
Derived forms: shacking up, shacked up, shacks up
Type of: dwell, inhabit, live, populate