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A couple who is now Catholic (or where one person is not but is willing to go through with such procedures for the sake of his/her spouse, children, etc.,), but has "marriage issues" can either go through the ceremony of matrimony or receive what is called a "Radical Sanation". Catholic Answers has an article about the latter.
Note that the article mentions "a dispensation for a marriage ceremony outside the Church". I don't know how often this occurs or what is involved (see, e.g., CCC, #1633-1637). It also talks about getting a radical sanation even without knowledge of one of the parties:
If your spouse would have an extremely bad reaction to the sanation procedure, then, for the sake of domestic peace, he would not need to be told about it: "A sanation can be granted validly even when one or both of the parties are unaware of it, but it is not to be granted except for serious reason" (CIC 1164). The extreme reaction of your spouse could count as the serious reason needed for this.This (if possible to obtain) is probably the best course for a Catholic married to a person who is quite opposed to Catholicism. Another EWTN page explains the difference between a convalidation and radical sanation (the latter makes the original vows retroactively valid and the former is a new set of valid vows). It also mentions one partner being able to do this:
Reasons for a radical sanation include a situation when one party will not cooperate in having the marriage convalidated through a ceremony . . .Apparently some priests have told couples they need not worry about such details of marriage convalidation. Another EWTN page (Fr. Mark J. Gantley) decisively answers this:
If a Catholic is married outside the Catholic Church, then the marriage must be convalidated. This is true whether the marriage involved a Catholic and a non-Catholic or two Catholics.The necessity of "ecclesiastical form" is discussed in CCC #1630-1631. This is my understanding of Church teaching. I'm not infallible, of course, but I have backed myself up with official Catholic sources, solid apologetics material, and priests from reputable and orthodox websites (I humbly submit that some priests who have been giving out goofy, erroneous information concerning these matters might do well to do the same, since this is a very serious matter).
For even more detail, one can consult the Code of Canon Law, that discusses marriage in its entires: #1055-1140. Here are a few of these:
Can. 1108 §1 Only those marriages are valid which are contracted in the presence of the local Ordinary or parish priest or of the priest or deacon delegated by either of them, who, in the presence of two witnesses, assists, in accordance however with the rules set out in the following canons, and without prejudice to the exceptions mentioned in canon 144, 1112 §1, 1116 and 1127 §23.
Can. 1117 The form prescribed above is to be observed if at least one of the parties contracting marriage was baptized in the catholic Church or received into it and has not by a formal act defected from it, without prejudice to the provisions of can. 1127 §2.
Can. 1118 §1 A marriage between Catholics, or between a catholic party and a baptized non-Catholic, is to be celebrated in the parish church. By permission of the local Ordinary or of the parish priest, it may be celebrated in another church or oratory.
§2 The local Ordinary can allow a marriage to be celebrated in another suitable place.
§3 A marriage between a catholic party and an unapprised party may be celebrated in a church or in another suitable place.