In general, what you describe ("multicellular organisms that can have tissues rip/split off and then rejoin later, not even necessarily in the same part of the organism") does not occur in the natural world, except maybe the mycelium or possible the plant example below.
Some trees can branch apart and them have separate branches merge back together, but they have not been separate organisms.
I suppose you could have a plant have a branch root near the parent, have the connection to the parent broken in some way (making it a separate organism), and then have their branches merge together as described above, but I have not heard of this.
There are however, lab made examples of cutting off body parts getting and putting them back together in ways unlike their original situation.
Classic hydra regeneration experiments, but also any studies that have used that technique to study different aspects of developmental biology. This would include frogs, chicks, fish, many insect species, and crayfish off the top of my head.
Some simple animals like a sponge can be dissociated to cells and them mushed back together. After a while they will reorganize into a recognizable and functioning sponge.
In the natural world, if a part of an animal were to split off and be able to rejoin at a later time, it would have to acting like an independent organism, which would make it seem to be an example of vegetative reproduction (making a new individual by budding off a mini-version of an adult). Many animals (like hydra) can do this, but they don't hang around to rejoin the parent. They have their own lives to live at this point.
There are also other examples of an animal "losing" a part as a defense or escape mechanism, like a lizard losing it tail or a crayfish losing a leg, if grabbed by a predator. These parts are then left to confuse or occupy the predator while the loser of the body part escapes.
There may be examples of a body part coming off an animal and being transferred to another individual as a part of reproduction (transferring a sperm packet), but I can't think of any specific examples right now.
At a super-organism level (like an ant colony), a part becoming separate from the original (super)-organism (the ant colony) become a separate colony and them merge together in some way (due loss of queen?), but that's pretty different from a single organism situation.
...sorry. 
