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Neither VMI nor the Citadel has Greek Life. ATO and Sigma Nu were founded at VMI, but fraternities were banned there early on. They have waaaaaaaaay too intense of a campus life to have time for Greek life (VMI is across the street from my college). I assume that the Citadel also bans Greek Life--there probably wouldn't be interest in it anyway.
Davidson College's eating houses do indeed function as sororities in some sense. There is also a co-ed eating house. They have discussed letting NPC sororities colonize the eating houses numerous times but it always gets rejected. The important difference in the eating houses is that there is a quota system, but selection is done by lottery. You choose your own house, and you can preference with a group of friends, as well. There is a lot of information about it on Davidson's Patterson Court website. (Patterson Court is the home of the fraternities and eating houses.)
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