Based on the basic idea of marrying through a messenger (Shaliah) both by a man and a woman (Kiddushin 41).
First, the bride and the groom say to the messengers: "go bring me any man/woman". So one messenger meets another and says to him before two witnesses: "let your sender (woman) be Mekudeshet to my sender (man) with this ring" and voila, the senders are married, having no idea of to whom (and the fact that they are married at all, but that's a different question).
How does this work? Why in Judaism is it possible to get married without knowing at all to whom?
- I'm aware of the Gemmorah ibid that Rav forbids marrying without seeing first, but that did not become an obligating Halachah.
- I'm aware of the idea of buying a cow without seeing, but as I understand Kinyan Ishut (Kinyan Issur) is (or should be) different (somehow).