  We show that a K\"ahler-Ricci soliton on a Fano manifold can always be
smoothly approximated by a sequence of relative anticanonically balanced
metrics, also called quantized K\"ahler-Ricci solitons. The proof uses an
equivariant version of Berezin-Toeplitz quantization to extend a strategy due
to Donaldson, and can be seen as the quantization of a method due to Tian and
Zhu, using quantized Futaki invariants as obstructions for quantized
K\"ahler-Ricci solitons. As a by-product, we show that a K\"ahler-Einstein Fano
manifold does not necessarily admit anticanonically balanced metrics in the
usual sense when its automorphism group is not discrete.
