“I actually don’t understand why the diocesan tribunal even deals with the SSPX-married couple if the marriage sacrament was without validity in the Church.”
To receive a writ (whatever the proper term is) saying exactly that. That way the person can marry again WITHIN the Church and with perfect validity.
“They wouldn’t deal with a civilly-married couple, would they (I have no knowledge or experience with this — correct me if wrong).”
Yes, to receive essentially the same sort of writ. I know someone who did so. He wanted to be able to marry within the Church and wanted it to be clearly understood that he never had a valid marriage previously.
The SSPX has gone so far as to pretend it has authority it clearly does not: http://www.sspxseminary.org/publications/rectors-letters-separator/rectors-letter/226.html
Looks like that would be "Declaration of Nullity".
I am still a little puzzled because I was told that the Church considers every marriage to be valid a priori, whether a Catholic Sacrament of Matrimony was involved or not. An annulment is obtained only if some defect is found in the previous marriage, even a civil marriage, such as one party having been coerced. Or maybe this is the old pre-Vatican II world and now anything goes as a reason?