When using the arsonist against the Hedge Tavern, where does the Hedge Tavern card go? Does it go to the bottom of an "Era of Turmoil" stack or does it go back to its original starting position face up?
I think they key phrases to consider here are:
Choose 1 of your opponent's buildings adjacent to a settlement/city. He must place it under a draw stack of his choice
--From the instructions on the Arsonist action card.
In each Theme Set, there are expansion cards that must be accessible to both players. Separate these cards and place them as a face-up expansion card stack next to the draw stacks. These cards are never part of your hand.
--From page 14 of the rule booklet.
Two of these could point to an exception - the arsonist sends a building to a draw stack - and the face up expansion card stack is next to the draw stacks - implying that this stack is perhaps not defined as a draw stack itself; also most cards explicitly state that a destroyed or discarded card is placed under a matching draw stack, where the Arsonist simply says one of his choice.
If we look at the rules explaining how the Fraternal Feud event card is resolved, we see:
... you either place them both at the bottom of the same stack or at the bottom of 2 different stacks. All your opponent knows is under which draw stack(s) you placed the cards.
--From page 20 of the rule booklet.
There can be a tactical advantage to deciding which draw stack a discarded card goes under, even in the basic version of the game where there are no expansion card or theme set draw stacks to choose: a small stack may allow a player to get that card again soon (or if there's an empty stack, the player who's turn it is may be able to return the card to their hand at the end of their turn) and a large stack will guarantee that the discarded card remains out of play for a longer period of time.
The Arsonist gives the choice of the stack to the victim. This limits the power of the card by not allowing the active player to effectively "steal" a card from her opponent's principality and add it to their hand. Or at least, not without some cooperation, or exceptionally few cards remaining in the stacks.
Of course, the Hedge Tavern is one of the face-up expansion cards, and as such it "must be accessible to both players." This is further ensured by limiting principalities to only having one at a time - so the other would be accessible to the other player.
Also, whenever there are possible implicit inconsistencies in a rule set, the best practice is to rule in favour of the mandated rule, rather than the implied one. Of the phrases highlighted above, only "never part of your hand" is mandated by its wording. If the Hedge Tavern were placed under one of the theme set draw stacks, this would enable it to appear in a player's hand later in the game.
It seems clear that:
The game designers did not intend the Hedge Tavern to end up in a player's hand.
The game designers intended the Hedge Tavern to always be available to both players when playing it's Theme Set - never permanently destroyed as an option, nor held hostage by the other player.
Ergo: There's not a lot of room on that bottom line on the card. Most likely they left out the word "matching" as it wouldn't fit and seemed somewhat redundant by the time you're playing the second theme. If the Arsonist is played, and the Hedge Tavern is destroyed, it returns to the face-up Expansion Card stack.
I would play as per the wording on the Arsonist card. Just because the Hedge Tavern has an unusual starting placement, in a face-up stack, doesn't mean it should be exempted from subsequent effects.
Given that there is an stack available with the same back as the Hedge Tavern, I see no reason why you shouldn't follow the course of action prescribed by the Arsonist card, and put the Hedge Tavern to the bottom of such a stack of the opponent's choice.