  The antiferromagnetic Ruddlesden-Popper ruthenate Ca$_3$Ru$_2$O$_7$ is a
model polar metal, combining inversion symmetry breaking with metallic
conductivity; however, its low temperature ($T < 48$ K) crystal structure and
Fermi surface topology remain ambiguous despite numerous measurements and
theoretical studies. Here we perform both first principles calculations with
static correlations and angle resolved photoelectron spectroscopy experiments
to construct a complete model of Ca$_3$Ru$_2$O$_7$, reconciling inconsistencies
among interpretations of electrical transport, thermopower measurements, and
momentum- and energy-resolved band dispersions. The solution relies on treating
the interplay among Coulomb repulsion, magnetic ordering, spin-orbit
interactions, and the RuO$_6$ octahedral degrees-of-freedom on equal footing.
For temperatures $30<T < 48$ K, we propose weak electron-electron interactions
produce a symmetry-preserving metal-semimetal transition with Weyl nodes in
proximity to the Fermi level, whereas a new orthorhombic $Pn2_1a$ structure
emerges for $T<30$ K, exhibiting charge and spin density waves from enhanced
Coulombic interactions.
