  We show that a collinear Heisenberg antiferromagnet, whose sublattice
symmetry is broken at the Hamiltonian level, becomes a fluctuation-induced
ferrimagnet at any finite temperature $T$ below the N\'eel temperature $T_{\rm
N}$. We demonstrate this using a layered variant of a square-lattice
$J_1$-$J_2$ model. Linear spin-wave theory is used to determine the
low-temperature behavior of the uniform magnetization, and non-linear
corrections are argued to yield a temperature-induced qualitative change of the
magnon spectrum. We then consider a layered Shastry-Sutherland model,
describing a frustrated arrangement of orthogonal dimers. This model displays
an antiferromagnetic phase for large intra-dimer couplings. A lattice
distortion which breaks the glide symmetry between the two types of dimers
corresponds to broken sublattice symmetry and hence gives rise to
ferrimagnetism. Given indications that such a distortion is present in the
material SrCu$_2$(BO$_3$)$_2$ under hydrostatic pressure, we suggest the
existence of a fluctuation-induced ferrimagnetic phase in pressurized
SrCu$_2$(BO$_3$)$_2$. We predict a non-monotonic behavior of the uniform
magnetization as function of temperature.
