  Chiral spin liquids (CSLs) in three dimensions and thermal phase transitions
to paramagnet are studied by unbiased Monte Carlo simulations. For an extension
of the Kitaev model to a three-dimensional tricoordinate network dubbed the
hypernonagon lattice, we derive low-energy effective models in two different
anisotropic limits. We show that the effective interactions between the
emergent $Z_2$ degrees of freedom called fluxes are unfrustrated in one limit,
while highly frustrated in the other. In both cases, we find a first-order
phase transition to the CSL, where both time-reversal and parity symmetries are
spontaneously broken. In the frustrated case, however, the CSL state is highly
exotic --- the flux configuration is subextensively degenerate while showing a
directional order with broken $C_3$ rotational symmetry. Our results provide
two contrasting archetypes of CSLs in three dimensions, both of which allow
approximation-free simulation for the thermodynamics.
