Hola, Yvonne.
Sugiero
(1) formalisation of de facto marriage, o
(2) conversion of de facto marriage in/into civil marriage

Como en Venezuela no se aplica el “common law”, yo diría “de facto marriage” en lugar de “common law marriage”; la parte de “regularizar” puede ser “formalise” o “convert in/into civil marriage”.

Sugiero “convert/conversion” basándome en que es un tipo de unión que, para efectos legales, se convierte en otro tipo; la referencia que doy abajo se refiere a uniones homosexuales, pero no creo que haya impedimento en usar el mismo verbo para uniones heterosexuales.

En años recientes el término más usado es “unión estable de hecho” en lugar de “concubinato”, aunque hay jurisprudencia del TSJ que establece que no toda unión estable de hecho es concubinato.

[CONCUBINATO]:
De las anteriores definiciones, podemos escudriñar los caracteres de la unión concubinaria que se distinguen, como bien lo establece el artículo 77 de nuestra Constitución, de otras uniones no matrimoniales o de hecho y que se caracteriza por:
• Notoriedad de la comunidad de vida.
• Unión monogámica entre individuos de sexo diferentes.
• Unión permanente.
• Ausencia de impedimento para contraer matrimonio.
• Desenvolvimiento de una vida intima semejante a la matrimonial.
• Inexistencia de las formalidades del matrimonio.
(http://guarico.tsj.gov.ve/decisiones/2005/octubre/350-11-577...
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[De Facto Marriage]:
De Facto Marriage Law and Legal Definition
A de facto marriage is a term used for certain couples who live together without being legally married. Recognition of de facto marriages varies by jurisdiction and entity. In some instances, an entity will recognize a de facto marriage, so that a person is eligible for the same rights as a legally married person. Some typical requirements are that a de facto spouse lives under the same roof as the member and that possessions have been pooled for joint use as a common household. It may also be referred to as "common-law union" or "cohabitation."
(https://definitions.uslegal.com/d/de-facto-marriage/)
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De facto marriage is a term used to define the relationship between a couple who is not legally married, yet meets the civil requirements for marriage. Whether the relationship is legally recognized as a marriage typically depends on the jurisdiction in which the couple resides. In the United States, for example, most states recognize de facto marriages, although such unions are now defined as domestic partnerships. Many jurisdictions in Europe and the United Kingdom, as well as Israel, Canada, and Australia, also recognize these types of informal marriages.
Latin for in fact or reality, the term de facto is most commonly used in legal contexts. It is applied to situations in which an act exists due to circumstance, yet has not been formally established by law. In terms of marriage, a de facto marriage exists when two individuals reside together and represent themselves as spouses to the outside world.
Frequently known as a common law marriage, de facto marriage is defined by the nature of the relationship, as well as the couple’s cohabitation activities. In other words, if two individuals live with one another for a specified period of time, they may be legally married in the eyes of the law — even if no actual marriage ceremony has taken place.

(https://www.wisegeek.com/what-is-a-de-facto-marriage.htm)

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[Conversion]
Court of Appeal upholds High Court judgment in civil partnership case
“Since the enactment of same-sex marriage, the retention of civil partnerships is no longer a necessity. However, same sex couples who have chosen not to convert their partnership to civil marriage should be allowed to remain in civil partnerships if they so choose.”
(https://www.familylaw.co.uk/news_and_comment/court-of-appeal...