  We use MUSE/VLT to conduct a survey of $z\sim3$ physical quasar pairs at
close separation with a fast observation strategy. Our aim is twofold: (i)
explore the Ly$\alpha$ glow around the faint-end of the quasar population; (ii)
take advantage of the combined illumination of a quasar pair to unveil
large-scale intergalactic structures extending between the two quasars. Here,
we report the results for a quasar pair ($z=3.020,3.008$; $i=21.84,22.15$),
separated by 11.6 arcsec (or 89 projected kpc). MUSE reveals filamentary
Ly$\alpha$ structures extending between the two quasars with an average surface
brightness of SB$_{\rm Ly\alpha}=1.8\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$
arcsec$^{-2}$. Photoionization models of the constraints in the Ly$\alpha$,
HeII, and CIV line emissions show that the emitting structures are
intergalactic bridges with an extent between $\sim89$ and up to $\sim600$ kpc.
Our models rule out the possibility that the structure extends for $\sim 2.9$
Mpc, i.e., the separation inferred from the uncertain systemic redshift
difference of the quasars if the difference was only due to the Hubble flow. At
the current spatial resolution and surface brightness limit, the average
projected width of an individual bridge is about 35 kpc. We also detect a
strong absorption in HI, NV, and CIV along the background sight-line at higher
$z$, which we interpret as due to at least two components of cool, metal
enriched, and relatively ionized CGM or IGM surrounding the quasar pair. Two
additional HI absorbers are detected along both quasar sight-lines at $\sim
-900$ and $-2800$ km s$^{-1}$ from the system, with the latter having
associated CIV absorption only along the foreground quasar sight-line. The
absence of galaxies in the MUSE field of view at the redshifts of these two
absorbers suggests that they trace large-scale structures or expanding shells
in front of the quasar pair.
