Why aren't all particles virtual? It is my understanding that virtual particles are those that are internal to a particular interaction. They do not fulfill the requirements of a freely-travelling particle (ie a plane wave) and as such do not have the usual rest mass of "real" particles. In other words, they are "off mass shell". Many sources mention that virtual particles should not be treated as particles at all, and instead describe them as strictly mathematical constructs to calculate the integrals represented by Feynman diagrams.
My confusion with this description is, what really makes something internal to an interaction? For any given "external" line in a Feynman diagram, why couldn't we simply expand the diagram to include further interactions of the outgoing particle, such that it becomes "internal" to our new diagram, and therefore a virtual particle? Unless we're talking about a universe of only a couple of particles that really do interact once and never again, I don't see a fundamental distinction between these two types of particles. People talk about how only non-virtual particles are measurable, but isn't any measurement just another long series of interactions, of which our "measured" particle is some internal part?
To give an extreme example, take a photon emitted from the sun by some process, which travels all the way to Earth before interacting with some other particle on the surface. Couldn't I construct a Feynman diagram of this interaction, where the photon is just an internal line exchanged between two other particles, making it "virtual"?
Have I made a fundamental misunderstanding of virtual particles here, or are all particles in QFT, real and virtual, simply excitations of the field that asymptotically approximate on-shell particles the further away the two interaction nodes are?

Edit: There seems to be some lack of consensus on this question. Some answers to similar questions are saying that the notion of being "on-shell" is simply an asymptotic approximation for very long-lived particles, and that fully on-shell states are unphysical as they correspond to a completely non-interacting particle. In other words, the difference between "real" and "virtual" particles is not a discrete distinction:
Does radio use virtual photons?
"Slightly off shell?"
Other answers seem to say that virtual particles are fundamentally different to real particles. They only make sense within the context of a Feynman diagram, and a Feynman diagram is strictly a useful tool for perturbative calculations. A single Feynman diagram does not make physical sense on its own, and does not represent the actual, physical world lines of particles:
Are W and Z bosons virtual or not? (see "Can one distinguish real and virtual photons?")
 A: There are a few things to clarify:
We use theories to model the real world, and our currently accepted theory is the SM, together with QM and GR. These theories are then tested experimentally and to our current knowledge, the SM model is the one that best fits the data from the experiments.
Real (elementary) particles are the elementary particles in the SM. These particles all show up in our experiments. Yes, even real neutrinos. You can have a real physical trace of all these real particles in an experiments. That is why we call them real particles. This is a real direct trace (in most cases).
Now, virtual particles are just a mathematical model of describing an interaction as a phenomenon, that exists between a field and another real particle.
The distinction was needed, because there was a problem when we tried to describe the interaction between a field and another real particle. You might get confused because, some start asking questions like, "How does the EM field of an electron interact with a proton?".
Then you try to answer this question, and you realize, in reality, we do not know how. All we know, is that the EM field of an electron has an effect on spacetime, so that in the region of spacetime where the field exists, then in that region, when a real (another) particle interacts with the EM field, the particle will have an altered trajectory. The field will have an effect on the particle, but this will be a indirect effect (on its trajectory).
Now in the case of gravity, it is more obvious. This effect is always attractive and acts on all known real particles in the SM. We use the phrase then "gravity bends spacetime". 
Then you ask the question "How?".
In reality we do not know. All we know, is that the field has an effect on the fabric of spacetime, and we try to model it. This mathematical model of this effect is called virtual particles. We do not know how it works, but we know the effect, based on the calculations in the model of virtual particles.
So basically the answer to your question is:


*

*real particles are real because they themselves do have a direct effect (trace) in our experiments. And yes, real neutrinos are real too, though it is very hard to experimentally see their direct trace since they interact weakly with normal matter.

*virtual particles are a mathematical model of the phenomenon called interaction with the field, and these virtual particles only have indirect effects on other particles' trajectories in the experiments. Virtual particles will never leave a trace for example in a bubble chamber experiment.
Ultimately, the answer to your question is the field. We do not know how fields interact, that is why we talk about virtual particles in the case of a field. The real reason for our lack of knowledge about the fields' interactions is why we call these particles virtual.
Maybe one day, when we figure out how gravity or electromagnetism (or the other forces) have an effect on spacetime (how they bend spacetime), or what it bends or how it bends it or what the fabric of spacetime really is, then we will be able to tell if a virtual particle is a real thing or not.
