  The intense photon fluxes from relativistic nuclei provide an opportunity to
study photonuclear interactions in ultraperipheral collisions. The measurement
of coherently photoproduced $\pi^+\pi^-\pi^+\pi^-$ final states in
ultraperipheral Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV is
presented for the first time. The cross section, d$\sigma$/d$y$, times the
branching ratio ($\rho\rightarrow \pi^+ \pi^+ \pi^- \pi^-$) is found to be
$47.8\pm2.3~\rm{(stat.)}\pm7.7~\rm{(syst.)}$ mb in the rapidity interval $|y| <
0.5$. The invariant mass distribution is not well described with a single
Breit-Wigner resonance. The production of two interfering resonances,
$\rho(1450)$ and $\rho(1700)$, provides a good description of the data. The
values of the masses ($m$) and widths ($\Gamma$) of the resonances extracted
from the fit are $m_{1}=1385\pm14~\rm{(stat.)}\pm3~\rm{(syst.)}$ MeV/$c^2$,
$\Gamma_{1}=431\pm36~\rm{(stat.)}\pm82~\rm{(syst.)}$ MeV/$c^2$,
$m_{2}=1663\pm13~\rm{(stat.)}\pm22~\rm{(syst.)}$ MeV/$c^2$ and $\Gamma_{2}=357
\pm31~\rm{(stat.)}\pm49~\rm{(syst.)}$ MeV/$c^2$, respectively. The measured
cross sections times the branching ratios are compared to recent theoretical
predictions.
