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Can I actually get a share of stock issued with a piece of paper anymore?
Yes you can. One additional "advantage" of getting the physical certificate is you can use it to transfer your account from one brokerage to another.  You get the certificates in the mail and then just send them to the new broker. Why anyone would want to go through this extra work (and usually added expense) rather than a direct transfer is beyond me but it is one additional "advantage" of physical certificates.