  We study the zero-temperature phase diagrams of Majorana-Hubbard models with
SO($N$) symmetry on two-dimensional honeycomb and $\pi$-flux square lattices,
using mean-field and renormalization group approaches. The models can be
understood as real counterparts of the SU($N$) Hubbard-Heisenberg models, and
may be realized in Abrikosov vortex phases of topological superconductors, or
in fractionalized phases of strongly-frustrated spin-orbital magnets. In the
weakly-interacting limit, the models feature stable and fully symmetric
Majorana semimetal phases. Increasing the interaction strength beyond a finite
threshold for large $N$, we find a direct transition towards dimerized phases,
which can be understood as staggered valence bond solid orders, in which part
of the lattice symmetry is spontaneously broken and the Majorana fermions
acquire a mass gap. For small to intermediate $N$, on the other hand, phases
with spontaneously broken SO($N$) symmetry, which can be understood as
generalized N\'eel antiferromagnets, may be stabilized. These antiferromagnetic
phases feature fully gapped fermion spectra for even $N$, but gapless Majorana
modes for odd $N$. While the transitions between Majorana semimetal and
dimerized phases are strongly first order, the transitions between Majorana
semimetal and antiferromagnetic phases are continuous for small $N \leq 3$ and
weakly first order for intermediate $N \geq 4$. The weakly-first-order nature
of the latter transitions arises from fixed-point annihilation in the
corresponding effective field theory, which contains a real symmetric tensorial
order parameter coupled to the gapless Majorana degrees of freedom, realizing
interesting examples of fluctuation-induced first-order transitions.
