  The HeII transverse proximity effect -- enhanced HeII Ly$\alpha$~transmission
in a background sightline caused by the ionizing radiation of a foreground
quasar -- offers a unique opportunity to probe the morphology of quasar-driven
HeII reionization. We conduct a comprehensive spectroscopic survey to find
$z\sim3$ quasars in the foreground of 22 background quasar sightlines with
HST/COS HeII Ly$\alpha$~transmission spectra. With our two-tiered survey
strategy, consisting of a deep pencil-beam survey and a shallow wide-field
survey, we discover 131 new quasars, which we complement with known SDSS/BOSS
quasars in our fields. Using a restricted sample of 66 foreground quasars with
inferred HeII photoionization rates greater than the expected UV background at
these redshifts ($\Gamma_\mathrm{QSO}^\mathrm{HeII} > 5 \times
10^{-16}\,\mathrm{s}^{-1}$) we perform the first statistical analysis of the
HeII transverse proximity effect. Our results show qualitative evidence for a
large object-to-object variance: among the four foreground quasars with the
highest $\Gamma_\mathrm{QSO}^\mathrm{HeII}$ only one (previously known) quasar
is associated with a significant HeII transmission spike. We perform a stacking
analysis to average down these fluctuations, and detect an excess in the
average HeII transmission near the foreground quasars at $3\sigma$
significance. This statistical evidence for the transverse proximity effect is
corroborated by a clear dependence of the signal strength on
$\Gamma_\mathrm{QSO}^\mathrm{HeII}$. Our detection places a purely geometrical
lower limit on the quasar lifetime of $t_\mathrm{Q} > 25\,\mathrm{Myr}$.
Improved modeling would additionally constrain quasar obscuration and the mean
free path of HeII-ionizing photons.
