I heard the following analogy when talking to some specialists in absolute de Rham theory. I think Deninger's name was mentioned at about the same time.

One possible way to imagine a variety over $\mathbb{F}_p$ is as a manifold equipped with a distinguished vector field, which we call "Frobenius". The usual discrete Frobenius that admits integer powers is the unit time evolution of the flow. Orbits of points on the variety that are defined over finite fields correspond to closed integral curves of the flow, and we assume that such curves only have integral periodicity.

There are a few problems. If there are points defined over an imperfect field, you may have to consider flows that start somewhere, like a distinguished submanifold. Also, it is not clear to me how one connects such a picture to the set of solutions of a system of polynomial equations.

**Edit:** (response to Daniel Litt's comment) I must confess that I am not particularly familiar with the idea, so I can only fill in vague guesses. Also, I don't know why one would want to interpolate between different powers of Frobenius. The fundamental idea seems to be that if we examine the spectrum of a finite field using étale glasses rather than Zariski or Nisnevich glasses, it looks a lot like a circle, since the étale fundamental group is a completion of $\mathbb{Z}$. This suggests that if we were to propose some real geometric object as an analogue of a variety, $\mathbb{F}_q$-points should be distinguished circles, perhaps with some distinguished automorphism.

The picture of finite fields as circles also shows up in "arithmetic topology" speculation for similar reasons. Here, the spectra of number rings are viewed as 3-manifolds, and the finite points are distinguished embedded circles, for which some kind of linking number may be defined homologically. As far as I know, this is another analogy that seems to be waiting for a substantial application.