how will this purchase be tracked? if you pay not by card or in cash, how do they arrange everything
Good question.
Well, the transaction will be recorded in the blockchain when you transfer your Btc to another address. So, that will serve as your proof of purchase. And of course it will be processed from a specific crypto payment processor before completing the signing of papers.
I'm not expert and haven't tried anything like this, but pretty sure these are some basic steps.
That's right, whenever we make a purchase we get the billing details. When we make a purchase with bitcoin, the transaction details along with the purchaser name gets added. Another thing it is clearly said by Elon Musk, those Bitcoins used by people on purchase won't be moved to fiat. It'll be stored in a separate vault. Therefore not much of transactions were involved. Checking a single transaction id itself everything can be connected with the purchaser.
If the buyer doesn't want his main wallet to be connected to the transaction, It would be a smart move to use third-party services like a bitcoin mixer so that the transaction can't be detected and can preserve the owner's identity. In real-life, Doing paperwork and putting up legitimate information is inevitable so the only thing you can put into private is between the wallet address and you.