If a woman goes through informal recruitment/COB, don't most schools still require them to sign a type of MRABA binding agreement once they accept a bid?
At IU, the COB-MRABA form is part of the recruitment materials given to the chapters.

The difference with that is if you sign an open bid,you obviously know what you are getting. When you sign your MRABA you could get one of three.

My understanding is that, if you accept a bid through COB, pledge, and then depledge, you cannot receive another bid for one calendar year (including the next FR). If you receive a bid but do not accept, you are not bound.

My experience has been that they do sign some paperwork for the Greek Life office so they can be added to that chapter's roll. Usually it has been a statement that they are accepting a bid from that chapter.

Here is how it works (I’ll try to skip commentary)

GCU holds formal rush in fall 2019. Zsa Zsa signs an open bid from ABC shortly after formal rush is completed.

It turns out ABC and Zsa Zsa are not a good fit so she depledges 3 weeks into her program. Zsa Zsa is eligible to rerush in fall 2020.

Magda didn’t go through formal, but met a lot of sisters in ABC in fall 2019. ABC is well under total and decides to have a pledge class in spring 2020. They give Magda a bid and she accepts, but ABC doesn’t work for her either and she depledges too.

However, per NPC rules she did not join through the “primary recruitment” at the school, and is allowed to go through formal rush in fall 2020, even though it has not been a whole calendar year.

I’m not sure how it works if you sign a bid to a colony and then quit.