The tape is represented by tokens whose strength increases with their distance from the read head, such that the current cell is the weakest one. On every loop iteration, it gets killed (read) and triggers different effects depending on its type, such as getting replaced (writing a new value), changing the effects that will be active on the next iteration (changing state) and dealing damage to one side of the tape but not the other (moving the read head).
I think what they do is set up a sequence of cards that 'initialises' the game state with various creatures and resources. These together with the game rules form the Turing machine. Then they have a sequence of cards that correspond to the 'paper tape' of symbols a Turing machine operates on.
> "Our result is also highly unusual in that all moves of both players are forced in the construction. This shows that even recognising who will win a game in which neither player has a non-trivial decision to make for the rest of the game is undecidable."
It's not so strange: the game is Turing complete, so the winner can be undecidable because who wins is the result of some arbitrarily complex computation.
Imagine we play a "game" where I win if there's a nontrivial zero of the Riemann zeta function with real part not equal to one half, otherwise you win. Neither of us has any decisions to make in this game. It's still very difficult to determine who's going to win.