Depends on what you mean by "go after."
A co-signer promises to answer for the debts of another. If the tenant doesn't pay the rent, the co-signer is obligated to pay. If neither pays, then the landlord will, presumably, eventually sue. Generally, the landlord will sue for breach of contract and obtain a money judgment. It would be extremely rare for a landlord to require a co-signer to give a deed of trust, but I doubt that it has never happened. If it did happen, then the landlord could foreclose. However, if the landlord only has a money judgment, then he/she/it can enforce that judgment in the ways provided in the Code of Civil Procedure. The most common ways of enforcing a civil money judgment are wage garnishment and bank levy. The judgment creditor can record an abstract of judgment, which places a lien against all real property now or in the future owned by the debtor in the county where the abstract is recorded. However, actively enforcing a civil money judgment against real property is uncommon and, in the case of a judgment debtor's personal residence, so rare as to not be worth considering.