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It's not realistically Turing complete. It assumes infinite precision.

Right but a Turing computer assumes infinite storage space which is itself impossible. You cannot have infinite precision without infinite storage, and all real computers that we colloquially say are Turing complete have finite everything.

It lazily uses infinite tape though, so it is Turing complete for an actual, finite number of steps. You can’t have infinite precision even for a single step.

It's possible to argue that there's no such a thing as infinite precision floats. I.e. all objects have limited "size", and real numbers is just a nice abstraction to simplify discrete world. (https://en.wikipedia.org/wiki/Finitism)

Turing machine is not such thing. At each moment in time, only finitely many cells of the tape are used. (The same applies to natural numbers, there are infinitely many of them, but each one of them has a finite description).

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