It's not so much keeping track of time, but that that time matches the real world outside the machine. The whole Tetris Complete formalization is about how a g

It's not so much keeping track of time, but that that time matches the real world outside the machine. The whole Tetris Complete formalization is about how a given machine (which as a previous comment points out, doesn't need to be turing complete) relates to the non-theoretical non-computational world. Turing completeness requires or even suggests any of those things, it's an orthogonal concept.