Wang tile turing machine tile placement
Any proof that the Wang tiling problem is undecidable by reduction from undecidability of the halting problem of Turing machines will contain instructions on how the Wang tiles simulate the operation of the machine, and indeed a proof that a tiling exists if and only if the machine doesn't halt. 

There is more than one such proof. If you look at different proofs, you will see different reductions. There is absolutely no reason to expect that different reductions correspond to the same tiling placements. Indeed, there is no canonical way of converting Turing machines to tiling problems. Therefore there is no "reliable algorithm" for the problem; any specific proof should include such an algorithm that works for the specific proof.

I suggest you try to look up a technical paper with such a proof. Perhaps you can find a university library with Berger's original paper, or one of the later papers cited in the Wikipedia article.