The KEY is for the anonymous buyer or Principal to purchase the property through what is often referred to as a nominee. Here are two scenarios. (1) Nominee enters into a contract to buy property and then assigns contract to Principal -- Principal, not nominee, closes on purchase or (2) Nominee takes title to the property and later conveys it to Principal.
What might be more important here is your reason for Anonymity and/or the circumstances surrounding the transaction. It is easy to perceive of what could go WRONG under such a situation. If your intent is to proceed with this purchase, it would be in your best interest to see the advice of an attorney.
See 5th question in LINK below:
A similar technique (and, in fact, it's compatible with Bruce's scenarios) is to form either an LLC or land trust. The offer is made in the name of the LLC or land trust. (You'd still have to apply for, or at least sign for, the financing; few lenders will actually lend to a newly formed LLC or land trust.) I'm not a lawyer, so this isn't legal advice. However, for privacy and other matters, I'd favor the land trust, though I know investors who are perfectly comfortable using an LLC.
The the purchase could be made in the name of the LLC or land trust. In the case of the land trust, you could keep your identify hidden for as long as you want.
Technically, you could keep the LLC or land trust in place. You then could lease the property from the land trust or LLC. Or, if you don't mind revealing your identify after the purchase, you could transfer the deed from that legal entity to yourself.
A good lawyer should be able to structure it in whichever way works best for you.
Hope that helps.
In many ways I feel it is best to be forthright from the beginning. If the seller finds out later that it is you they might feel wronged somehow and bad feelings are never good for a transaction.
Minnesota requires an earnest money check (probably has your name and address on it), and in addtion the MAR approved pruchase agreement as for the buyers name on the 2nd line. If you told us a little more about why you want to be annonymous an agent here might have a great idea of how to accomplish your goals.