I thought Phi Beta Kappa was an academic honors club or society, where students at elite schools would be automatically signed up if they fell in the top (mumble mumble) percentage of their class.
Not a fraternity, nothing in the least secretive, not Skull and Bones. More similar to getting a "Summa Cum Laude" on your degree, but styled as an honors club instead of a degree credential.
Any sauce to what you have?