  We establish the unique ergodicity of a fully discrete scheme for monotone
SPDEs with polynomial growth drift and bounded diffusion coefficients driven by
multiplicative white noise. The main ingredient of our method depends on the
satisfaction of a Lyapunov condition followed by a uniform moments' estimate,
combined with the irreducibility and the strong Feller property for full
discretization. We transform the original stochastic equation into an
equivalent random equation where the discrete stochastic convolutions are
uniformly controlled to derive the desired uniform moments' estimate. Applying
the main result to the stochastic Allen--Cahn equation driven by multiplicative
white noise indicates that this full discretization is uniquely ergodic for any
interface thickness. Numerical experiments validate our theoretical results.
