Common law marriage was ended in South Carolina July 24, 2019.  This only applies prospectively.  In other words, all claims of common law marriage that existed prior to July 24, 2019 are still valid claims.

But South Carolina prohibited same sex marriage until Obergefell v. Hodges, 576 U.S. 644, 675–76 (2015).  In Obergefell the Supreme Court held "same sex couples may exercise the fundamental right to marry," and all state laws to the contrary were "invalid to the extent they exclude same sex couples...".

Since common law marriage between same sex couples was illegal in SC until 2015, can a person claim that a same sex, common law marriage was begun prior to 2015?

This issue was directly addressed on November 10, 2021 by the South Carolina Supreme Court in  Swicegood v Thompson .  The Supreme Court reversed the lower courts that found the law existing at the time Swicegood alleged the formation of a common law marriage barred them from being married, therefor no common law marriage could ever have existed.

The Supreme Court found this ruling was erroneous.  The court ruled that "Our state's restriction on same sex marriage in section 20-1-15 was rendered void ab initio by Obergefell and, therefore, must be treated as though it never existed."

Some statutes are revised/amended or withdrawn, but the impact of the prior statutes are still valid and enforceable.  The difference with Swicegood is that the South Carolina statute was ruled to be unconstitutional as a violation of equal protection of laws.  In this case, the rules require that we pretend the prior law never existed (void ab initio- from the beginning).

This is a significant ruling. It means same sex couples are free to allege common law marriage that existed prior to July 24, 2019, and are subject to all the rights and obligations of marriage... and divorce.