  Many seemingly contradictory experimental findings concerning the
superconducting state in Sr$_2$RuO$_4$ can be accounted for on the basis of a
conjectured accidental degeneracy between two patterns of pairing that are
unrelated to each other under the $(D_{4h})$ symmetry of the crystal: a
$d_{x^2-y^2}$-wave $(B_{1g})$ and a $g_{xy(x^2-y^2)}$-wave $(A_{2g})$
superconducting state. In this paper, we propose a generic multi-band model in
which the $g$-wave pairing involving the $xz$ and $yz$ orbitals arises from
second-nearest-neighbor interactions. Even if time-reversal symmetry is broken
in a $d+ig$ state, such a superconductor remains gapless with a Bogoliubov
Fermi surface that approximates a (vertical) line node. The model gives rise to
a strain-dependent splitting between the critical temperature $T_c$ and the
time-reversal symmetry-breaking temperature $T_\text{trsb}$ that is
qualitatively similar to some of the experimental observations in
Sr$_2$RuO$_4$.
