  Using unbiased Monte Carlo simulations and variational analysis, we present
the ground state and finite temperature phase diagrams of an exactly solvable
spin-orbital model with Kitaev-type interactions on a square lattice. We show
that an array of new gapped and gapless vison crystals -- characterized by the
periodic arrangement of $\mathbb{Z}_2$ flux excitations -- can be stabilized as
a function of external magnetic field and exchange anisotropy. In particular,
we discover a variety of `quarter phases' wherein new sixteen-site periodic
patterns emerge, with only a quarter of the fluxes adopting 0-flux
configurations. In contrast, the rest remain in $\pi$-flux configurations.
Vison crystals break translational symmetry and undergo finite temperature
phase transitions. We investigate the finite temperature properties of these
phases and report the corresponding critical and crossover temperatures. Our
results reveal an array of novel phases in exactly solvable extensions of the
Kitaev model, wherein local and topological orders can coexist.
