[antepodius]

It's a data visualisation of a property of a model of hydrogen. The dots are denser where the absolute value of the 'wavefunction' (which is just a function that takes in space and time coordinates and returns a complex value) is higher.

It's not 'real' in the same way a simulation of a tennis ball flying through the air, rendered with dots isn't, but worse: that's a visualisation of a model, too, but if you imagine it being a simulation of possible sensory data it makes more sense- the world is set up so it's possible for a tennis ball to be seen by a conscious being, but that isn't true for a hydrogen atom.

In a model of a tennis ball, you might have a black screen, and then draw a dot where the tennis ball is, according to the model.

In this model of the hydrogen atom, you have a black screen, and then you draw a cloud with denser and less dense regions to represent where the electron 'is', according to the model. The problem is that electrons aren't point particles; in this model, an electron is a cloud- it's described not by some vector describing its position, but by the aforementioned wavefunction. It's a cloud in space, (except every point is complex-valued- they're taking the magnitude for this rendering) that changes (or doesn't) over time.

There's layers here. To what degree is a simplified model 'real'? To what degree is a visualisation of a model a picture of a 'real' thing, even if that model were true and complete?

[throwaway_pdp09]

I wasn't clear - sorry. It obviously can't be seen, and I (just about) get it's a probability cloud. My question is, is the underlying probability cloud real, in the sense that we can think of it existing and in it's actual peculiar shape, so if we could probe it we would indeed find something shaped like that, or is it just an abstraction/model 'that just works'?

It's not even an easy question to ask, come to think of it.

[mncharity]

As commented elsewhere, the OP renders look perhaps not quite right. But in general...

Atoms are little balls. With electron density that's mostly spherically symmetric. It's very high at the nucleus, and falls off exponentially outward. Down by several orders of magnitude by the time you reach distances at which atoms hang out together.

A 2D analogue might be a stereotypic volcano, if height were density. I wish I knew of a better one. Few familiar objects have this degree of fuzzy. Diffusing smells, but you can't see those.

The common representations with a solidish surface at some large distance from the nucleus is... useful when doing chemistry, but badly misrepresents the physical object.

Electron density manifests clearly and concretely. For example, you can poke at it with the vibrating tip of an Atomic Force Microscope.

Electron states, orbitals, seem less often encountered that directly. Rather than at one step remove - seeing density, or some other phenomena, and explaining it with states.

Though there's a fun STM image I'm just now failing to quickly find. An STM scans a tip across a sample, measuring and mapping the tunneling current between them. Usually with a boringly symmetric ball of a tip, so the interestingness is all sample. In this case however, the sample had a grid of boring s states, and tip conduction was through a tilted f state. So the sample repeatedly mapped the tip. The image is thus a grid of little images of the tip's f state, laid out like rows of little buoys. EDIT: Maybe this is it: https://arxiv.org/pdf/cond-mat/0305103.pdf fig 6 page 12 (though it doesn't entirely match what I'm remembering).

Just to be clear, representing density by dots, rather than say by a color gradient on voxels, is purely a data-visualization rendering choice. Like old newspapers using halftone images. Which I mention only because there are misconceptions around the individual dots themselves representing something about the electron.

[cinntaile]

Yes, the probability cloud tells you how the hydrogen atom (H-O-H) will be shaped. The atoms will be repelled by the electrons that create this probability cloud and this will force the atoms into a geometry with minimal energy (which is the shape you are probably familiar with). Just remember that it's a probability so the shape isn't set in stone, it's more like the most common shape. A common way to learn this is by implementing the Hartree-Fock algorithm. https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method

[throwaway_pdp09]

Well how about that, ask a difficult question of an abstract domain, and get two clear yesses. Does not happen often. Thanks @mncharity and @cinntaile.