I am very sympathetic to Gisin and his cause, but he does not propose any sensible resolution. By the way, not a fault, and no blame for him. Pointing out logical deficiencies always comes before a satisfying solution, and he is to be praised for his insight.
There are many interesting ways to probe this problem.... here's one:
Say I tell you to imagine a circle, an ideal Platonic circle in a Cartesian coordinate system (real coordinates, first uneasiness). Let's ignore translation, so it is centered at (0,0). I tell you the radius. Can you imagine the circle with Plato? Model the circle? Reproduce the circle? Do you need pi? Does the circle include or encode pi? But pi is has infinite information.
Perhaps all you need is the square root function? But that's also an infinite Taylor series expansion. You can plot and recreate the circle to any precision if you have a square root function. The series will only need to run to the required precision. The circle will always be granular, depending on the number of terms you use in pi, or the square root function. Yeah, right, obvious, so why is that a problem?
What if I tell you the circle is the physical manifestation of equipotentials of a stationary charge (say, nucleus), or mass (say Earth), with inverse square law - so basically a geometric fall-off with range determined by spatial (circular, spherical) considerations. What is the force at some distant point? Do you need pi? Do you need square root function? Or reciprocals? How does the other charge or mass feel the 56,323rd decimal place of the force due to the potential?
Maybe it doesn't, because by the time it has felt the second decimal place, time has moved on, the charges/masses have moved on, and the nuance of what would've/should've been felt in a never changing universe are never experienced. There is a modified differential equation that relates various time derivatives to precision of experienced forces (this almost sounds like relativity :)
The discrete explanation with photons goes like this: the force is produced by radiating photons. They automatically encode the geometric expansion as inverse square law, because of their pathways, no need for pi, or sqrt functions. But that is statistical, the accuracy is only as good as the number of photons that can arrive from the source. The circular/spherical nature of the force only emerges over time, as photons arrive and act. The accuracy of smooth circularity and inverse square only establishes itself over time...
Elapsed time affects experienced precision - hmmm, interesting.
How would you quantify such a thing, where time changes the precision of what you feel? Well, the other obvious example is the Heisenberg Uncertainty Principle. This is just a simple and obvious example of Fourier Analysis for any theory based on a linear wave equation. It almost doesn't need stating, and if it must have a name, it is certainly Fourier, not Heisenberg. Anyway, any math/physics/engineering student knows Fourier to their core, and it gives a nice solution to the information problem: coordinates may be real-valued degrees of freedom, but there is no way to mathematically or physically resolve all coordinates and their derivatives to infinite precision. It's just not possible, even if the underlying equations/reality maintain the fiction of real-valuedness.
Fourier combines time, waves, amplitude, velocity (momentum, etc.) with a specific expression for possible information. A picture is worth a thousand words at this point, just look at a wave-packet, it's obvious. Fourier is a masterwork, and vastly underappreciated as a fundamental limit on knowledge, in a real world sitting on smooth continuous waves.
So Fourier sets limits on knowledge, even in the wavy world of the smooth continuum. Of course, I do not believe in the smooth continuum anyway, but Fourier is my wingman to fight the real-infinitists on their own smooth turf.