A single, stable particle is always on-shell. Thus your claim is trivially correct if there is nothing else around. But this is a physically dull situation.
More interesting is already an electron in an external electromagnetic field.
In quantum field theory, perturbation theory produces in lowest order a sum of infinitely many tree diagrams with comb-like trees having a stem representing the electron and wiggly teeth representing soft photons. The free ends are real particles, everything else is virtual.
An interaction vertex with an onshell electron and an onshell photon must result in an offshell electron because of energy conservation, which is strictly enforced in Feynman diagrams. (There is no room in the formalism for a lack of energy conservation that would enable virtual particles to exist with "borrowed energy", as wikipedia would like to have it!.)
2a. If diagrams of quantum field theory are taken wholly serious (and interpreted in the meaningless but popular space-time fashion), the electron is on-shell at time $-\infty$ and time $-\infty$, but sligthly of-shell at all finite times, as it interacts with an infinitude of on-shell photons representing the electromagnetic field, and an interaction vertex with an onshell electron and an onshell photon must result in an offshell electron because of energy conservation (which is strictly enforced in Feynman diagrams).
2b. But if particles are called real if they have a state that changes with time then an electron in an external field is always real and on-shell (as off-shell states do not even exist formally). This is consistent with the usual quantum mechanical description of the electron, and this description can be justified from quantum field theory in the appropriate semiclassical approximation (the e/m field being generated by many far away particles).
With two electrons, the only dynamically valid description is again in terms of real electrons only (as in the nonrelativistic case). In contrast, the standard Feynman diagrams with their internal lines aka virtual particles only describe the scattering situation (real particles at time $\pm\infty$), with complete disregard of the details that happen at finite times.