  We examine the class of compact Hermitian manifolds with constant holomorphic
sectional curvature. Such manifolds are conjectured to be K\"ahler (hence a
complex space form) when the constant is non-zero and Chern flat (hence a
quotient of a complex Lie group) when the constant is zero. The conjecture is
known in complex dimension two but open in higher dimensions. In this paper, we
establish a partial solution in complex dimension three by proving that any
compact Hermitian threefold with zero real bisectional curvature must be Chern
flat. Real bisectional curvature is a curvature notion introduced by Xiaokui
Yang and the second named author in 2019, generalizing holomorphic sectional
curvature. It is equivalent to the latter in the K\"ahler case and is slightly
stronger in general.
