What are the legal issues with a buyer remaining anonymous with an offer to purchase real estate?
The largest concern I would have with anyone being anonymous in a contract for the purchase and sale of Real Estate would be performance of and under the contract.
Pending your state, a purchase agreement has specific stages where all parties need to perform specific steps towards the eventual closing. If there is anonymity in the agreement, there cannot be recourse if the contract steps are not properly executed. In many circumstances, the buyer will likely need to obtain financing, or if cash is to be paid for the property, they will need to provide proof of availability of funds to actually close the transaction. Proof needs to be supplied that the borrower is able to obtain financing, or has sufficient capital to acquire the property and anonimity will not be possible. A seller will also incur costs to move a property to closing with title search, counsel fees and other typical closing costs for selling a parcel. If the buyer is unknown there is no recourse (recovery options) for the seller other than the buyer posting a performance bond. Legally, if you need to sue for specific performance under the contract, who would you sue?
You should consult with a real estate attorney before proceeding.