If you haven't seen Shadow's dreams, you're probably wondering why this sections is here -- after all, Shadow and Relm don't exactly interact all that often. But if you have seen the dreams, you know that one of the biggest mysteries of Final Fantasy VI is whether or not Relm is Shadow's daughter. Shadow never tells the party point blank that Relm is his daughter (in fact, he never tells them anything regarding his past), but there are many things that point to this conclusion. Let's take a look at them:
Okay, so we've all seen Shadow's dream -- but have you seen Relm's dream? If you choose not to save Shadow on the Floating Continent (shame on you!), you will see Relm's dream instead of Shadow's final dream, after you recruit her at the Cave in the Veldt. In the dream, you see Relm in a house in Thamasa, along with Strago and Interceptor. She asks Strago where "Daddy" is, and if he's coming back. Afterwards, you see Interceptor run out the door (which matches up perfectly with Shadow's final dream, in which you see Interceptor rush out the door of the house to join Shadow). If you put two and two together, it only makes sense that Shadow/Clyde is Relm's "Daddy." This is by far the most concrete evidence offered in the game.
This relic is described in the menu as follows: "Departed mom's love protects against fatal magic attacks." What does this have to do with Shadow, you ask? He and Relm are the only ones who are able to equip the ring -- why would Shadow be able to equip Relm's mother's ring and no one else? Not even Strago is able to equip it. This could be totally random, but I think it's a pretty legitimate piece of evidence.
If you talk to all of the townsfolk in Thamasa, you can learn a few bits of important information about Strago. The most important bit of information comes from a woman in a shop. She tells you, "Actually, Relm isn't Strago's real grandchild. I heard she's his friend's daughter." While this seems to have nothing to do with Shadow, it pieces together nicely with everything else we've learned -- most importantly, it explains why Strago can't equip the Memento ring -- he is not technically related to Shadow, Relm, or Relm's mother. Additionally, if you watch carefully during scenes with Strago and Shadow, Strago seems to be familiar with Shadow. It's very subtle, but it's there.
Like I said, none of these things prove that Shadow is Relm's father, but I think it's the only legitimate explanation. There are too many pieces of evidence for it to be a mere coincidence, and nothing else seems to fit. Some people argue that Relm is Shadow's sister (because of the Memento ring's description), but I don't think that fits well with the dreams. Other people argue that Shadow can't be her father because he never comes forward and tells her the truth, but I don't think that a legitimate argument, either. After all, Shadow left Relm because he felt inadequate -- he wouldn't have left if he thought being in her life could benefit her, so there's no point in disrupting her life after she got used to the idea of living with Strago. Take it or leave it, but that's just my two cents!