  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.
