Note: even if you do move your shares, the brokers regularly take them back, so check regularly
The client would need to either sign the back of their certificate held in the vault or sign something known as a "stock power" which is attached to the certificate and takes the place of signing the cert. It's either of these methods that would make the certificate "negotiable" and allow it to be taken onto the firm's books and out of the client's name.
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