  The symmetry requirement and the origin of magnetic orders coexisting with
superconductivity have been strongly debated issues of iron-based
superconductors (FeSCs). Observation of C$_4$-symmetric antiferromagnetism in
violation of the inter-band nesting condition of spin-density waves in
superconducting ground state will require significant change in our
understanding of the mechanism of FeSC. The superconducting material
Sr$_2$VO$_3$FeAs, a bulk version of monolayer FeSC in contact with a perovskite
layer with its magnetism (T$_N$ ~ 50 K) and superconductivity (T$_c$ ~ 37 K)
coexisting at parent state, has no reported structural orthorhombic distortion
and thus makes a perfect system to look for theoretically expected C$_4$
magnetisms. Based on variable temperature spin-polarized scanning tunneling
microscopy (SPSTM) with newly discovered imaging mechanism that removes the
static surface reconstruction (SR) pattern by fluctuating it rapidly with
spin-polarized tunneling current, we could visualize underlying C$_4$ symmetric
(2$\times$2) magnetic domains and its phase domain walls. We find that this
magnetic order is perfectly consistent with the plaquette antiferromagnetic
order in tetragonal Fe spin lattice expected from theories based on the
Heisenberg exchange interaction of local Fe moments and the quantum order by
disorder. The inconsistency of its modulation Q vectors from the nesting
condition also implies that the nesting-based C$_2$ symmetric magnetism is not
a unique prerequisite of high-T$_c$ FeSC. Furthermore, the plaquette
antiferromagnetic domain wall dynamics under the influence of small spin torque
effect of spin-polarized tunneling current are shown to be consistent with
theoretical simulation based on the extended Landau-Lifshitz-Gilbert equation.
