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CzeV343 (=V849 Aur) was previously identified as a candidate double eclipsing binary (2+2 quadruple), where the orbital periods of the two eclipsing binaries ($P_A \approx 1.2$ days and $P_B \approx 0.8$ days) lie very close to 3:2 resonance. Here, we analyze 11 years of ground-based photometry, 4 sectors of TESS 2-minute and full-frame photometry, and two optical spectra. We construct a global model of our photometry, including apsidal motion of binary A and light-travel time effect (LTTE) of the mutual outer orbit, and explore the parameter space with Markov Chain Monte Carlo. We estimate component masses for binary A ($1.8+1.3 M_\odot$) and binary B ($1.4+1.2 M_\odot$). We identify pseudo-synchronous rotation signal of binary A in TESS photometry. We detect apsidal motion in binary A with a period of about 33 years, which is fully explained by tidal and rotational contributions of stars aligned with the orbit. The mutual orbit has a period of about 1450 days and eccentricity of about 0.7. The LTTE amplitude is small, which points to low inclination of the outer orbit and a high degree of misalignment with the inner orbits. We find that when apsidal motion and mutual orbit are taken into account the orbital period resonance is exact to within $10^{-5}$ cycles/day. Many properties of CzeV343 are not compatible with requirements of the 3:2 resonance capture theory for coplanar orbits. Future evolution of CzeV343 can lead to mergers, triple common envelope, double white dwarf binaries, or a Type Ia supernova. More complex evolutionary pathways will likely arise from dynamical instability caused by orbital expansion when either of the binaries undergoes mass transfer. This instability has not been so far explored in 2+2 quadruples.