What sort of theoretical machine would be needed to solve the tiling problem? So theoretically, what sort of machine would we need to solve the tiling problem?

(Given a set of tiles, decide if they will tile the plane or not.)

I know we could have a Turing machine plus a "tiling oracle", but do we know what, specifically a Turing machine isn't able to do that keeps us from being able to compute the answer?
According to Wikipedia, the halting problem reduces to the tiling problem. This means that one thing that prevents Turing machines from solving the tiling problem is their inability of deciding whether a given procedure ever halts. The tiling problem might be even more difficult that the halting problem, but unfortunately I wasn't able to confirm or disprove this.