Hi
Let me chip in something here.
Reading this thread reminds me of my own marriage certificate! Since the day I signed the paper with my wife, we have kept it in some secure drawer in our home. We never bring it out to check or remind ourselves that we are married because that's not what the certificate is for. (Actually, the certificate has tremendous benefits for us as marriage couple in our country but that's not the purpose of this thread so I won't elaborate.)
Been married > 18 years. We stay married because we commit the marriage to God. And more than that, we live as married couple (not as live in partner because there's a difference in the eyes of God). And if we encounter problems in the marriage (which probably run into hundreds or thousands over the last 18 years), we did not rely on the cert to stay together but on God's love and grace and favor.
That said, God's institution of marriage can also be integrated into a country's marriage system, provided the system does not go against the Word of God. And if the system offers benefits for registered couples, why not go along with it? However, I can't speak for every country in this.
And if you want to be a married couple without the certificate but recognized as married couple in the eyes of God, by all means. But you may (I think) run into problems with the law of your country. May God gives everyone wisdom in this issue.