sole-executive agreement, negotiated and entered into through the president's authority as commander-in-chief of the armed forces.
That is a different thing than a treaty.
According to the Congressional Research Service the United States has SOFAs with 47 other countries that are not based any treaty or underlying Congressional action. (making them sole-executive agreements)
http://www.fas.org/sgp/crs/natsec/RL34531.pdfSole-executive agreements do not have to be ratified by Congress and they are considered constitutional by the Supreme Court.
Examples of other sole-executive agreements include the Yalta Agreement of 1945, the Vietnam Peace Agreement of 1973, and the Iran Hostage Agreement of 1981.