When attempting to pick up a space containing a Refined Storage system, all disks are lost when the structure is replaced. Additionally, Grids lose their orientation.

Bags of Yurting can't implicitly handle loading/unloading blockentities that store positional or rotational data, or store data outside of the actual blockentity -- the other mod would need to explicitly tell Bag of Yurting how to handle these blockentities via the API (I provide an API for other mods to add support but I can't add the support myself as Refined Storage's developers understand how their data is stored much better than I would)

In the meantime, the server can add the relevant blocks to the bagofyurting:blacklist block tag to prevent players from damaging their buildings or inventories.

Well, that was fast. The Refined Storage devs don't seem interested in adding integration. Leaving the issue open because it technically isn't resolved, but a resolution doesn't seem apparent. If possible, it might be prudent to check if Refined Storage is loaded and add anything from it to the blacklist by default.