Generally if you look at the Feynman diagram for an interaction, you'll have some number of external legs which one thinks of as physical particles, as well as a number of interactions. The lines between the interactions may be particles whose mass exceeds the total amount of energy of the incoming particles, so they obviously can't be real. They are thus generally called virtual (or off-shell particles). How one can think of this is what I tried to address in the original response.
I appreciate you saying the external legs can be thought of as physical particles, since they are not technically physical particles. The renormalization groups tells us that those look like physical particles. Internally to the interaction this is not true, which is why they would be called virtual particles. But I see them as a pure mathematical relic of the perturbation calculation. Perhaps I am just not able to extend my intuition here. And likely a more depth description from you or someone else is not really practical here. I just come back to the fact that in the end, particles are just different excitation states of the system. There is a vacuum state, a bunch of states we would consider one particle states, a bunch more we would consider two particle state, and so on. Doesn't this cover all configurations of the system and there are no "in-between" or what you would call off-shell states? I think these only make sense when we are making an approximation around the non-interactive model.
