What is the proper way of documenting and reporting the loan when it is joint between a husband and wife and the primary borrower passes away?

What are you reporting?

Just want to be sure if we obtain a death certificate that we can change the ownership to just the spouse on our system with SSN if they wish. This will be used for IRS reporting for the 1098.

Also, just want to be clear that in order to totally remove the deceased borrower we wouldn't have to refinance.

Bump

Do I have this in the wrong forum?

I was waiting to see if someone else had first hand knowledge.
Let me state that this is my OPINION based on experience only.
Ownership is not determined by anything other than a recorded document, certainly not by adjusting your system.
The property passes to the spouse through probate procedures and the filing of a deed.
From a mortgage standpoint, I've never seen anything other than a refinance done to remove a deceased borrower.
You need to qualify the remainder spouse for the loan, not just remove the decedant.

So I can't just obtain a death cert to move the co-borrower to the primary for IRS reporting purposes?

No, I don't believe you can.
These issues are handled through probate and estate income tax filings.
As for your loan, how can you make a person a primary or sole borrower without making sure they qualify?

No, I am not talking about the note. This is secondary market and as long as the spouse continues to live in the property they assume the liability of the mortgage. A refinance is the only way to remove them from title.

My question was totally only for our customer information and IRS reporting.