When common law marriages are enforceable, in a situation where three or more individuals live together as a "married" romantic group, what determines the married or unmarried status of the individuals?
In jurisdictions that recognize common law marriage, it is generally necessary to show that the spouses agreed to marry each other, that they held themselves out to the public as married, and that they cohabited or consummated the marriage. Once this happens, a common law marriage comes into being.
In Colorado and most jurisdictions that recognize common law marriage, once a common law marriage is entered into it is binding for all purposes, and cannot be dissolved except by divorce.
In all jurisdictions that recognize common law marriage, any subsequent relationship which would otherwise constitute a common law marriage does not give rise to a common law marriage of someone who is already in a common law marriage, because the law does not recognize that you can be married to more than one person at once (and indeed, criminalizes marrying more than one person at once).
Many jurisdictions that do not themselves have common law marriage will recognize a common law marriage which was effective where it was entered into between the parties.
Colorado and some other states and some programs in the federal government (including the Social Security Administration) also recognize the rights of someone who is a "putative spouse" which is someone who believes in good faith that they are married to their spouse and that their spouse was not married to anyone else at the time that they believed that they were married to their spouse.
For example, someone who married in a state that requires a marriage license to do so, who thought that a marriage license was obtained when it wasn't, might be a putative spouse.
Similarly, someone who obtained a marriage license or met the requirements of a common law marriage at a time that their supposed spouse was divorced when in fact a decree of divorce was never obtained, might be a putative spouse. One spouse can be a putative spouse, even if the other one knows that the marriage isn't valid.
But, anyone who knew that their supposed spouse was already married at the time that they purportedly got married would not qualify as a putative spouse, because the person could not believe in good faith that they were married under a legal regime mandating monogamy.
Note that I am using the term "common law marriage" in the somewhat narrow sense of a marriage not formally licensed by the government based upon the principles of the English common law as received by the jurisdiction in question.
There are countries, in which, by statute or legal custom that does not derive from the English common law, marriages are recognized without being registered with the government. For example, a country might recognize Islamic law as effective for family law matters between Muslims, and might hold that a marriage not licensed by a government marriage license was still valid if the marriage was effected in a manner recognized by Islamic law. In that case, the first four unregistered marriages of a man made without a death or divorce of a spouse would be valid, and any further purported marriages would be invalid. But, a purported marriage of a married woman to another man would be void and probably punishable by some very severe Islamic law sanction.
The law of the jurisdiction in which the group is.
For example, Australia does not allow or recognise polygamous marriages but it recognises each of the participants as a de-facto partner for most applications of the law. If someone is from a state which does allow polygamy, only the first marriage is recognised in Australia as a marriage - the others are de-facto relationships.