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Platonists think there's a realm where concept exist, but that realm isn't physical. I am wondering if the possibility of various other physical realms existing was ever proposed and if it was included in some sort of grand metaphysical theory or theory related to mysticism.

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    How "other" is the physical realm supposed to be? Epicurus had multiple worlds with gods living in intermundial spaces. How about Everett's parallel universes or Bostrom's simulators?
    – Conifold
    Jul 16 at 23:30
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    The notion of a "realm" of Platonic Ideals is just a metaphor. Jul 17 at 6:17
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If you're specifically interested in other physical realms that aren't part of the same space that we inhabit (i.e. you couldn't get there by traveling some distance in space), this article talks about how a French bishop named Etienne Tempier argued in 1277 that Aristotle was wrong to argue that the ground under our feet had to be a unique collection of the element earth (one of the four basic elements believed in by Aristotelians) at the center of the "world", and that this prompted others medieval thinkers to consider possibility of other "worlds" not spatially connected to ours:

Among the ideas that Tempier condemned was a principle of Aristotelian thought that held that the “first cause” (or, as medieval scholars would have said, God) could not have made more than one world. The logic went something like this: Earth was among the world’s four key elements, and one of its principles was that it moved towards the center of the world. If there were a neighboring world to ours, though, with earth at its center, that earth wouldn’t be moving towards the center of our world. Since that violated the rules of how earth behaved, there could only be one world.

To Tempier, though, this idea went against a key theological principle: God was all-powerful and could accomplish whatever he willed. Since there couldn’t be limits on God’s power, there could be multiple worlds, if he wanted to make them.

Some medieval thinkers took this as a challenge. “Immediately they started to say, ‘Let’s look harder at what Aristotle said,’” says Clemens. They started to look more closely, for instance, at previous Aramaic comments on Aristotle and considered what else might be possible. “They found new ideas that were outside the bounds of the Aristotelian physics of the day,” Clemens says.

Richard of Middleton, for instance, who lived in the second half of the 13th century, responded to Tempier by affirming that it could be possible to have more than one universe: “God could have and could still now create another universe.” He tried to reconcile this with Aristotelian thought by arguing that the matter of a second world would stay in its own separate universe, and earth elements would gather at the center of each.

A later scholar, William of Ware, developed this idea further. What did it mean to talk about another world, he wondered? He didn’t think it was possible to have two neighboring universes: by definition, the universe should include all the creatures ever made. So how could there be more than one? He argued instead that multiple worlds would have to be entirely separate, with no way of interacting—what today we might think of as parallel universes.

“That’s the way we think of multiverses today,” Clemens says in his talk. “We think, in the modern parlance, that they’re causally disconnected spaces that cannot interact.”

The article is based on a talk by astronomer Christopher Clemens which can be seen here:

https://www.youtube.com/watch?v=yE93hO9dXaQ

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Many early concepts of cosmology were quite finite; the (probably flat) Earth was surrounded by one thing and another on all sides, with as often as not a big lid, the Biblical "firmament", clapped over us. God's realm (and maybe also Hells of one kind or another) lay beyond. But many were pretty hazy about the scientific physicality of such Heavens, Hells, Purgatories and the like; the distinction between "physical" and "other" was not really a thing in those days.

Eastern mystics had a less finite view. At least some branches of Hinduism and Buddhism conceived an infinitude of (what tends to get variously translated as) Universes, worlds or planets, on which various life forms share their cycles of birth and rebirth with ours and the others in an endless cycle of suffering. The goal of the mystic was to attain enlightenment, or oneness with deity, and thus end one's own cycle.

Modern parallel-worlds and brane theories are primarily the domain of physicists, though where the physical speculation ends and the philosophical speculation begins is debatable. According to the Everett parallel-worlds model, there are countless parallel physical worlds which have split off from ours and steadily diverged at some point in the past, and this process continues on as gargantuan a scale as ever. According to Witten's M-theory, our Universe is a four-to-ten dimensional hypermembrane, or brane for short, floating in an 11- or 12-dimensional plenum. There may be many other branes, of varying dimensionality, also floating around in the plenum.

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