A lot of folks don’t seem to catch the “treaties” portion of Article VI. Be wary my friends. Be very wary. (Try not to sound like elmer fudd when saying that.)
A sensible person would argue (correctly) both that order matters, so that the Constitution takes precedence over treaties in case of a conflict, and that treaties made outside the Authority of the United States are explicitly excluded by the qualifier "Treaties made, or which shall be made, under the Authority of the United States". The problem is that many who want power will pretend those issues do not matter, and we cannot trust even the Supreme Court to protect the Constitution.