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We need a digital representation of real numbers. One possible choice is: a real number r is represented as the source code of a function which takes a natural number n to the nth digit of r. This seems no more like “cheating” (and to me, it is less like cheating) than to represent a real number r by a line segment of length r.

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That can't represent most real numbers, although I don't know that we have actually established that physical lengths can.

Both representations have the feature that it's impossible to check whether two real numbers are equal.

Actually with the physical objects you could use one to occlude the other and test for light passage

Not with infinite precision.

Importantly, this is actually true of the real numbers.

I don't think I believe in "actual" real numbers.

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