WASHINGTON — In a pair of unsigned opinions, the Supreme Court on Monday restored the rights of an adoptive mother who had split with her lesbian partner and reversed a murder conviction tainted by prosecutorial misconduct.

The adoption ruling reversed one by the Alabama Supreme Court, which had refused to recognize the woman’s adoptions of three children, which had been granted by a Georgia court in 2007.

The woman, identified in court papers as V.L., said she was overjoyed.

“I have been my children’s mother in every way for their whole lives,’’ she said in a statement. “I thought that adopting them meant that we would be able to be together always.’’

The US Supreme Court’s opinion, which was unsigned and had no noted dissents, said the Alabama court had violated the Constitution’s “full faith and credit’’ clause. “A state may not disregard the judgment of a sister state because it disagrees with the reasoning underlying the judgment or deems it to be wrong on the merits,’’ the opinion said.

The two women in the case were in a committed relationship that started in 1995 and lasted about 17 years. They shared a last name.

One of them, identified in court papers as E.L., gave birth to a child in 2002 and to twins in 2004, both times by insemination from an anonymous donor. They raised the children together in Alabama until they broke up in 2011, and the adoptive mother, V.L., continued to see the children for a time.

When a dispute about the visits arose, V.L. turned to an Alabama court, which granted her visitation rights based on the Georgia adoption judgment. The Alabama Supreme Court reversed that, saying in an unsigned opinion that the Georgia judgment was not entitled to the “full faith and credit’’ ordinarily required by the Constitution “to the public acts, records and judicial proceedings of every other state.’’

The Alabama Supreme Court reasoned that the Georgia court had misunderstood Georgia law in allowing the adoption, saying that “Georgia law makes no provision for a non-spouse to adopt a child without first terminating the parental rights of the current parents.’’

In a second unsigned opinion on Monday, the Supreme Court reversed the murder conviction of Michael Wearry, who is on death row in Louisiana for the 1998 killing of Eric Walber, a youth who had been delivering pizzas. Prosecutors presented no physical evidence tying Wearry to the crime, instead largely relying on testimony from two men whose accounts were open to doubt.

“The state’s trial evidence resembles a house of cards,’’ the Supreme Court’s opinion said.