  The magnetic and electronic properties of the layered Zintl-phase compound
EuAl$_2$Ge$_2$ crystallizing in the trigonal CaAl$_2$Si$_2$-type structure are
reported. Our neutron-diffraction measurements show that EuAl$_2$Ge$_2$
undergoes A-type antiferromagnetic (AFM) ordering below $T_{\rm N} =
27.5(5)$~K, with the Eu moments (Eu$^{2+}$, $S = 7/2$) aligned
ferromagnetically in the $ab$ plane. The $H = 0$ magnetic structure consists of
trigonal AFM domains associated with $ab$-plane magnetic anisotropy and a
field-induced reorientation of the Eu spins in the domains is evident at $T =
2$~K below the critical field $H_{c1} = 2.5(1)$ kOe. Electrical resistivity and
ARPES measurements show that EuAl$_2$Ge$_2$ is metallic both above and below
$T_{\rm N}$. In the AFM phase, we directly observe folded bands in ARPES due to
the doubling of the magnetic unit cell along the $c$ axis with an enhancement
of quasiparticle weight due to the complex change in the coupling between the
magnetic moments and itinerant electrons on cooling below $T_{\rm N}$. The
observed electronic structure is well reproduced by first-principle
calculations, which also predict the presence of nontrivial electronic states
near the Fermi level in the AFM phase with $Z_2$ topological numbers 1;(000).
