In the case of Reid v. Covert (354 U.S. 1 (1957)), a US woman killed her soldier husband on a US military base in the UK. Pursuant to the Status of Forces agreement the US has with the UK, she was tried for murder in a military tribunal (which didn't have a jury). The US Supreme Court overturned the conviction, saying that Congress's powers, including the power to regulate the armed forces, were still limited by the Bill of Rights, including Trial by Jury.
The ruling in this case basically said that no agreement with a foreign nation can confer power on Congress, or any other branch of Government, which is free from the restraints of the Constitution.
The Court felt that Article VI, clause 2 was written the way it was written not to make Treaties supreme but instead to simple keep treaties written prior to ratification of the Constitution in force.
The Court also distinguished this case from Missouri v. Holland (252 U.S. 416 (1920)) by saying that in this case, the Treaty directly violated a Constitutional protection. In Holland, it was more of an indirect violation.