  In this paper, we study geometry of totally real minimal surfaces in the
complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the
harmonic sequence generated by these minimal immersions. For totally real flat
surfaces that are minimal in both $Q_{N-2}$ and $\mathbb{C}P^{N-1}$, we
determine them for $N=4, 5, 6$, and give a classification theorem when they are
Clifford solutions.
