Does the Catholic Church perform non-civil marriages in the Dominican Republic? In 2013, as many as 200,000 people born in the Dominican Republic were stripped of their citizenship, due to their Haitian ancestry.  Most of these people were thus rendered "stateless," making government courts unavailable to them.  Furthermore, many are destitute and find it virtually impossible to get proper legal identification, which often requires a trip to Haiti and high legal expenses.  (For more on the situation, see this 30-minute documentary or its 2-minute trailer)

From a religious perspective, one of the biggest challenges these people face is that they cannot be legally married – civil marriage is not available to them unless they have proper identification.  In at least some Protestant denominations, such people who faithfully live together and have children are thus considered to be committing adultery, and are not given the opportunity to have a "church wedding" to change this status.

I'd like to know how the Catholic Church handles this situation.  Put simply: does the Catholic Church offer "religious-only" (non-civil) weddings in the DR to those who cannot obtain a civil marriage?  If so, are these weddings considered sufficient by the Catholic Church to comply with biblical prohibitions of adultery and fornication?
The relevant passages of the Code of Canon Law are (emphasis mine):

Can.  1067 The conference of bishops is to establish norms about the examination of spouses and about the marriage banns or other opportune means to accomplish the investigations necessary before marriage. After these norms have been diligently observed, the pastor can proceed to assist at the marriage.
Can.  1071 §1. Except in a case of necessity, a person is not to assist without the permission of the local ordinary at:
1/ a marriage of transients;
2/ a marriage which cannot be recognized or celebrated according to the norm of civil law;
3/ a marriage of a person who is bound by natural obligations toward another party or children arising from a previous union;
4/ a marriage of a person who has notoriously rejected the Catholic faith;
5/ a marriage of a person who is under a censure;
6/ a marriage of a minor child when the parents are unaware or reasonably opposed;
7/ a marriage to be entered into through a proxy as mentioned in can. 1105.
§2. The local ordinary is not to grant permission to assist at the marriage of a person who has notoriously rejected the Catholic faith unless the norms mentioned in can. 1125 have been observed with necessary adaptation.

In principle, the people you mention fall into the second case of subsection 1 of canon 1071; which means that the pastor of the people wishing to marry would have to contact his local Ordinary (that is, his Bishop) to obtain permission to assist at the marriage. I don't expect such a permission is difficult to obtain, given the concrete situation, but it does show that &quot;a marriage which cannot be recognized or celebrated according to the norm of civil law&quot; can still be celebrated by the Church, given the pastor takes a bit of care to prepare beforehand.
So I recommend that those people who do not have any documents to talk to their pastors, and they will advise them on how to proceed in order to contract a valid marriage.