We used the Multi-Unit Spectroscopic Explore (MUSE) on the Very Large Telescope (VLT) to conduct a survey of z ∼ 3 physical quasar pairs at close separation (<30″) with a fast observation strategy (45 min on source). Our aim is twofold: (i) to explore the Lyα glow around the faint-end of the quasar population; and (ii) to take advantage of the combined illumination of a quasar pair to unveil large-scale intergalactic structures (if any) extending between the two quasars. In this work we report the results for the quasar pair SDSS J113502.03−022110.9 – SDSS J113502.50−022120.1 (z = 3.020, 3.008; i = 21.84, 22.15), separated by 11.6″ (or 89 projected kpc). MUSE reveals filamentary Lyα structures extending between the two quasars with an average surface brightness of SB_Lyα = 1.8 × 10⁻¹⁸ erg s⁻¹ cm⁻² arcsec⁻². Photoionization models of the constraints in the Lyα, He IIλ1640, and C IVλ1548 line emissions show that the emitting structures are intergalactic bridges with an extent between ∼89 kpc, the quasars’ projected distance, and up to ∼600 kpc. Our models rule out the possibility that the structure extends for ∼2.9 Mpc, that is, 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 ∼35 kpc. We also detect one strong absorption in H I, N V, and C IV along the background sight-line at higher z, which we interpret to be due to at least two components of cool (T ∼ 10⁴ K), metal enriched (Z > 0.3 Z_⊙), and relatively ionized circumgalactic or intergalactic gas surrounding the quasar pair. Two additional H I absorbers are detected along both quasar sight-lines at ∼−900 and −2800 km s⁻¹ from the system; the latter has associated C IV 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. Combining longer exposures and higher spectral resolution when targeting similar quasar pairs has the potential to firmly constrain the physical properties of gas in large-scale intergalactic structures.