I guess what I am envisioning would be a front-end client for Open Transactions,
which associates a bitcoin address with the shares,
allowing the issuer to for example, send dividends or coupon payments to share holders.
Here's how you would do that on OT:
1. Have a Bitcoin issuer, who holds actual Bitcoins, and who issues Bitcoin UNITS onto the OT server, where OT users can open BTC Unit accounts, and trade around these units (exchanging them back out with the Bitcoin issuer, whenever they want to get back off the server.) The long-term vision is to store the backing Bitcoins in a voting pool on the blockchain (rather than directly with an issuer) in order to protect the users from being ripped off by the issuer. Voting pools aren't coded yet though, so for now, you would have to trust a Bitcoin issuer, the same as a gold issuer or any other kind of issuer.
2. Have a stock issuer for each type of shares. For example, if you issued "Pepsi shares" onto the server, then you must be the Pepsi
issuer. You could issue some Pepsi shares, in that example, and then sell them on the OT market in return for BTC units, or any other currency issued there.
3. Whenever the Pepsi issuer is ready to pay a dividend, he chooses the currency it's paid in
(let's say, BTC units, or centicoins or millicoins) and he also chooses the
payout per share. OT will then loop through all the Pepsi accounts and send a cheque to each account holder, based on the number of shares in his account.
4. Each payee receives a dividend cheque in his inbox, denominated in BTC units. He can cash the cheque, or deposit it into any BTC-unit denominated account. (This is all working already.)
5. To get actual Bitcoins back off of the OT server, the payee would take his BTC units to the BTC units issuer, and get a blockchain transfer of actual coins.
6. It would be possible to code the OT server
itself to act as the BTC issuer, which would automate things without involving any third parties. But then you would have to trust the OT server with your Bitcoins during the time it is holding them, which means you are vulnerable to a malicious server, or a hacked server. I have therefore avoided this in order to protect the OT brand, preferring instead to aim for the eventual "voting pool on the blockchain" solution, which is more secure than having to trust an issuer.
It wouldn't be hard to write some scripts that receive Bitcoins (using whatever tools you would use for those already) and then use an OT-Script to issue units onto the OT server, or to remove said units, using whatever rules you devise based on what you are trying to do on the blockchain. The entire OT API is available in script form now, so there's no reason you can't automate behaviors using cron and existing Bitcoin command-line tools.
In other words, while OT does enable you to issue shares, and to pay dividends, it doesn't pay those dividends directly in BTC to a blockchain address. Instead, it pays the dividends in OT-BTC-units, internal to OT, by sending a cashier's cheque to each shareholder, just as it might pay dividends in gold, or silver, or other currencies. (Any OT units will work.) Payees would have to then bail their BTC-units back off the OT server (and back onto the blockchain), through their BTC issuer. (And eventually, through a multisig-based voting pool on the blockchain, which would act as the "issuer.")