The way it was working 12 months ago (with the newly introduced electronic lodgement system), was as follows:
1) Obtain all the documents needed from your home country that others have described above, which does NOT mean opening a bank account in Argentina, because, of course, that is impossible until you have concluded the process successfully and have your residency.
2) Once in Argentina, but before uploading the documents to the system, go to an ATM and withdraw an amount from the overseas bank account where the income from you trust or rental property is deposited.
3) Retain the ATM receipt and take a photograph of it.
4) The photograph of the receipt is one of the documents you upload (the system has a specific space for it).
You have now completed the loop. You have an income stream from an overseas source (the documents you bring from your home country and upload, once apostilled there and translated here, demonstrate this). You have evidence that this income reaches an overseas bank in the necessary amount (the documents you bring from your home country and upload, once apostilled there and translated here, must include several months of bank statements that show the money going in regularly). You have evidence (in the form of the photographed ATM receipt) that you can withdraw money from that overseas account here in Argentina.
A few weeks after lodging all the documents online, you will get an email inviting you to come into Migrations in person (the very next morning) to present all the original documents. If all goes well, they will issue the Precaria that day. If that Precaria ever turns into an actual residency, you will then be able to open your local bank account and started transferring the money from your overseas account into that local account, and thus comply with the conditions of the Rentista residency.
However, you will never get to the last step in the process. As others have said, nobody at Migraciones has the fainted idea about the Rentista category. They will tell you anything, and it will almost certainly be wrong. They will just keep reissuing your Precaria every 90 days and even if you put in a request to expedite the tramite, it will continue to go unattended. After a while, you will realize that they don’t know what to do with your documentation, but that it really doesn't matter; as long as they keep renewing your Precaria, you are living happily in Argentina.
The only catch is that in theory you can renew the Precaria every 90 days from the comfort of your home by pressing a button on the system. That, unfortunately, only works 50% of the time. If, 10 days after you pressed the button, the system still hasn’t generated the new Precaria, you will have to join the queue at Migraciones from 5 am to have them do it manually in building C. Go on the 11th calendar day; go any earlier and your 3 hours of queuing in the cold will have been in vain.
Bear in mind too, that since the recent changes to the taxation legislation, the asset overseas that you use to generate your income is, technically, subject to the annual wealth tax, at rate that could reach as high as 2.25%. If you are a risk taker, listen to those who argue that (i) AFIP aren’t interested in foreigners, and/or (ii) aren’t equipped to trace foreigners’ wealth, and/or (iii) that even though the “overseas domicile” loophole has been closed, a smart lawyer will be able to convince a judge in the future that it is still open. If you are not a risk taker, think very carefully about declaring an asset to one part of government (Migraciones), who could be asked to report it any time in the future to another part (AFIP—the tax collectors).