  In copper- and iron-based unconventional superconductors, the Bogoliubov
quasiparticles interact with a spin resonance at momentum $(\pi,\pi)$. This
interaction is revealed by specific signatures in the quasiparticle
spectroscopies, like kinks in photoemission and dips in tunneling. We study
these signatures, as they appear inside and around a vortex core in the local
density of states (LDOS), a property accessible experimentally by scanning
tunneling spectroscopy. Our model retains the whole nonlocal structure of the
self-energy in space and time and is therefore not amenable to a Hamiltonian
treatment using Bogoliubov-de Gennes equations. The interaction with the spin
resonance does not suppress the zero-bias peak at the vortex center, although
it reduces its spectral weight; neither does it smear out the vortex LDOS, but
rather it adds structure to it. Some of the signatures we find may have been
already measured in FeSe, but remained unnoticed. We compare the LDOS as a
function of both energy and position with and without coupling to the spin
resonance and observe, in particular, that the quasiparticle interference
patterns around the vortex are strongly damped by the coupling. We study in
detail the transfer of spectral weight induced both locally and globally by the
interaction and also by the formation of the vortex. Finally, we introduce a
new way of imaging the quasiparticles in real space, which combines locality
and momentum-space sensitivity. This approach allows one to access
quasiparticle properties that are not contained in the LDOS.
