Each tab is its own copy of Alderhoard. They all share one save in this browser, but a tab reads that save only when it loads, and writes its own copy over it every time it saves.
Gold
Gold you win in a cabinet is held as tickets until you walk out. The clerk turns them into gold, and the valley saves it within half a minute (✓ SAVED flashes at the top of the screen).
Your other tabs don't see that gold until you refresh them. Run of Five's refresh arrow shows the gold in its own tab, not the newest save.
The last tab to save wins. If an older tab saves after a newer one, its gold replaces yours, and so do its buildings, chests and keys.
Close or refresh a tab while you're inside a cabinet and the tickets you were holding are gone.
Stonebreaker and THE WARLORD
Stonebreaker reads your solved walls when you walk into it. Solve your twentieth wall in one tab and THE WARLORD stays locked in the other until you leave Stonebreaker and walk back in, or refresh.
A tab that hasn't caught up can undo walls. If it solves a wall, it saves its own shorter list over the longer one.
Contraption
Yards solved in another tab show up after a refresh, and the same undoing can happen.
The safe way
Play in one tab. To switch, walk out of any cabinet, wait for ✓ SAVED, close that tab, then refresh the one you're keeping.