Sub-Neptunes with planetary radii of Rp ≃ 2─4 R⊕ are the most common planets around solar-type stars in short-period (P < 100 d) orbits. It is still unclear, however, what their most likely composition is, that is whether they are predominantly gas dwarfs or water worlds. The sub-Neptunes orbiting bright host stars are very valuable because they are suitable for atmospheric characterization, which can break the well-known degeneracy in planet composition from the planet bulk density, when combined with a precise and accurate mass measurement. Here we report on the characterization of the sub-Neptune TOI-5789 c, which transits in front of the bright (V = 7.3 mag and Ks = 5.35 mag) and magnetically inactive K1 V dwarf HIP 99452 every 12.93 days, thanks to TESS photometry and 141 high-precision radial velocities obtained with the HARPS-N spectrograph. We find that its radius, mass, and bulk density are Rc = 2.86−0.15+0.18 R⊕, Mc = 5.00 ± 0.50 M⊕, and ρc = 1.16 ± 0.23 g cm−3, respectively, and we show that TOI-5789 c is a promising target for atmosp-heric characterization with both JWST and, in the future, Ariel. By analyzing the HARPS-N radial velocities with different tools, we also detected three additional non-transiting planets, namely TOI-5789 b, d, and e, with orbital periods and minimum masses of Pb = 2.76 d, Mb sin i = 2.12 ± 0.28 M⊕, Pd = 29.6 d, Md sin i = 4.29 ± 0.68 M⊕, and Pe = 63.0 d, Me sin i = 11.61 ± 0.97 M⊕. TOI- 5789 is a mutually inclined system as the difference between the orbital inclinations of planets b and c must be higher than ~4 deg. Nevertheless, from sensitivity studies based on both the HARPS-N and archival HIRES radial-velocity measurements, we can exclude the possibility that these relatively high mutual inclinations are due to the perturbation by an outer gaseous giant planet.