As you correctly note, a proton can't spontaneously rearrange its "real" and "virtual" quarks to create two protons and an antiproton. This would require an input of energy, since in the rest frame of the original proton you'd go from a total energy of $m_p c^2$ (the rest energy of the proton) to something greater than $3 m_p c^2$ (the rest energy of two protons and an anti-proton, plus whatever kinetic energy the resulting protons have.) If it can't happen when the proton is at rest, then by relativistic invariance, it can't happen in any other rest frame either.
Instead, you need to add energy to the proton somehow, usually by smashing another particle into it. For example, Segre & Chamberlain's discovery of the anti-proton was accomplished by colliding two beams of protons with each other:
$$
p + p \to p + p + p + \bar{p}.
$$
You can view this as one proton giving a bunch of energy to the quarks inside the other proton, and allowing them to rearrange themselves into two protons plus an antiproton. (It is, of course, a little more complicated than that in reality—identical particles and all that—but that's the basic idea.)
Similarly, the very existence of quarks was first confirmed by deep inelastic scattering experiments, in which an electron or other lepton is collided with a proton or other baryon:
$$
e + p \to e + \text{lots of hadrons}
$$
The electron transfers some of its energy to the real or virtual quarks inside the proton, which is enough to "shatter" the original proton into multiple other hadrons in the way you describe. It turns out to be much more likely that the resulting hadrons are mesons ($\pi, \rho, K$, etc.) rather than baryons such as the proton or neutron, but it's certainly possible that "lots of hadrons" could be something like $p + p + \bar{p}$.
