Legal Question in Real Estate Law in New York

Yes. They must foreclose and obtain a judgment against you.

If I read your question correctly, you and your wife are both on title to the property but you are the only borrower on the loan. If this is the case, as an co-owner to the property, your wife must have signed the security instrument (mortgage) acknowledging that if you default on the loan, the lender can obtain and judgment against you and foreclose on the entire property taking her ownership interest as well.

I suggest you speak with an attorney if you are delinquent in your loan payments and are potentially facing foreclosure.