Ok, I searched, but didn't find anything, so sorry if I double post this.
Stacked background items are not keeping their order, when I quit and come back, or will be in a different order for players.
I find that if I use the "Send to Front/Send to Back" menus it's much more reliable, but still not 100% effective, especially if I have multiple overlapping items.
This screenshot shows a segment of arched alcoves which drops under one of the tiles. I used the "Send to front" command in the right click menu, then connected as a player, and it's still under the tile.
I suspect, but I'm not sure that it relates to using the "send to front" and "send to back" commands on multiple overlapping objects, since I also sent the piece of floor to the front, so it would overlap one of the adjacent floor tiles (the tiles come with included walls, and previously I had a rogue wall just hanging out in the middle of the room)
That series of ordering seemed to have stuck, but it appears when two objects "In front" collide, it's anyone's guess who gets to be the front one.
Of course, this is just all my best guess at what's going wrong. I also haven't had any coffee this morning, so it's entirely possible that I sound like the guy who tells the computer tech "My printer won't print, so I tried deleting some stuff from My Documents to speed up the internet"
TL;DR
Z-axis ordering is not persistent when I quit, and will also display differently for players.