I asked the professor of liturgics here about this question, and, although we were pressed for time so I couldn't get a full answer, he said that there are actually two different kinds of cloths. One of them, the Antimension, is consecrated during the consecration of an altar, signed by the Bishop and should have relics sown into it. Another -- whose name I cannot remember right now!! -- is just a piece of cloth with the Bishop's signature. Strictly speaking, one needs the full-fledged Antimension only when serving on a non-consecrated altar. If serving on a consecrated altar, however, one can use the "secondary" cloth with just the Bishop's signature. That way, one has (a) the holy relics in the altar, and (b) the Bishops' signature/approval, without having to have a full-fledged Antimension (which would redundant!).
I grilled him a bit about Antimensia because, in my experience, I have not seen many Antimensia with actual relics sown into them. Obviously, I haven't touched the Antimensia, but I do look at them carefully as the priest unfolds them, and many seem to me to just be the secondary version, i.e. the cloth with the signature, despite the fact that the Church altar is not consecrated. (Especially ones I have seen in several OCA and Antiochian parishes). I'm not sure about this, but such seems to be the case. Perhaps they are Antimensia in the sense that they have been consecrated and rubbed on relics, but there aren't enough bits of relic to go around.