What is the moral status of common-law marriage? It is a common Christian belief that sex outside of marriage (whether extra-marital or pre-marital) is sinful. But what about common law marriage? In 10 US states and the District of Columbia (source), a marriage can be contracted by a cohabiting couple putting themselves forward as married - beyond mere cohabitation, there is a necessity to present one's selves as being married. (Which as a practical matter is not hard - many people will assume that a cohabiting couple are married, so in a common-law marriage jurisdiction, a cohabiting couple can become married simply by doing nothing to disabuse others of the perception that they are married.) So, if a couple have a legally recognised common law marriage, are they sinning by living together and having sexual relations? Is a marriage ceremony or a legal document morally necessarily to be married? (If it is, did Adam and Eve have a ceremony or a contract?)

Does the jurisdiction in which the couple live matter for the purposes of its moral status? Is a cohabiting couple, presenting as married, in a common law jurisdiction in which such presentation is sufficient to become married, not sinning by having sexual relations, while a cohabiting couple, making the same presentation, in a jurisdiction which does not recognise common-law marriage, sinning? (Many jurisdictions which used to recognise common-law marriage no longer do - that is true of most US states, most Canadian provinces, Scotland, among others.)

If the legal status of the relationship is relevant to its moral status - what is the moral relevance of jurisdictions (like Australia or France) which provide legal recognition to cohabiting couples which is legally distinct from marriage? (i.e. de facto status in Australia, pacte civil de solidarité in France). If legal recognition is morally significant, does it morally matter whether a jurisdiction considers a relationship established by repute to be equal in status to formal marriages, or to have some distinct legally recognised status?

(In this question I am only asking about heterosexual couples who are exclusively monogamous, who are freely consenting, adults, of sound mind, and not related.)
Allowances of the law have little or no bearing on the moral standing of marriage or any other Christian practice. Where applicable we are bound to follow relevant laws in addition to the prescription of our doctrine, but this is in addition to not in place of them. Common law does not define marriage, although it may or may not make an applicable legal hoop necessary.

Most of marriage practice among Christians boils down to tradition, but the traditions are built on principles. While much variation is found in the details, the principles guiding what is or is not done are pretty fixed. Marriage is a covenant relationship between one man and one woman entered into by profession before men and sealed by God.

How you go about that, the pieces have to all be there.  Where the profession is made -- in church or on a lawn, whether officiated by an uncle or pastor or even not at all -- doesn't matter nearly so much that a declaration is made public that two people are now married. Where they sign state paperwork before our after our not at all if not required doesn't change anything. Making such a declaration, then sleeping with someone else would be a violation of that covenant no matter where the state stands.

Christianity even honors marriage as sacred when entered into by two non Christians with a purely secular profession. God's role in the covenant relationship is something we believe just is -- even if the parties involved ignore him.

Obeying the law of the land is required, but our duties do not end where the law stops.