Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) of an interaction in quantum field theory.
No virtual particle physically appears in the interaction: all possible virtual particles, and their antiparticles, occur equally and together in the mathematics, and must be removed by integration over the values of their momenta.
In the coordinate-space representation of a Feynman diagram, the virtual particles are on-mass-shell (realistic), but only 3-momentum is conserved at each vertex, not 4-momentum, so there is no immediate way of obtaining 4-momentum-conserving delta functions.
In the momentum-space representation, the virtual particles are both on- and off-mass-shell (unrealistic), but 4-momentum is conserved at each vertex, and also round each loop (as shown by a delta function for each).
In the coordinate-space representation, each virtual particle appears “as itself”, but in the momentum-space...