At my school, it's generally unspoken big/lil sister/brother. I have both a little and big brother, so do other members of my chapter. It is "frowned upon" by our nationals because of the sterotypes that come along with it (for example, when my big's national consultant came, he didn't completly understand how big/little was run at our school, in other chapters, they are just drinking buddies or sex partners).
My big and I became close and he asked me to be his little one day, same goes for my little, he asked me to be his big. My big's big sis is from a totally different chapter than I am. You generally have one big sis/brother from your own chapter and one from a fraternity/sorority, then as many littles as you want/can afford. The chapter never assigns you a specific chapter to be with, let alone your big/little.
Typically, the bigs get the littles a hoodie with the LITTLE's letters on it (you can't wear another chapters letters unless you go through certain ceremonys as detailed in bylaws or ritual) and underneath it will have the big's chapter SPELLED OUT lil' bro/sis. The littles usually make a paddle. I made my big a paddle and letters, same for my little.
I love my big and little, and don't really see how the "tradition" could become so skewed like you guys describe
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