Money illusion is believed to be instrumental in the Friedmanian version of the Phillips curve. Actually money illusion is not enough to explain the mechanism underlying this Phillips curve. It requires two additional assumptions.
First, prices respond differently to modified demand conditions: an increased aggregate demand exerts its influence on commodity prices sooner than it does on labour market prices. Therefore the drop in unemployment is, after all, the result of decreasing real wages and an accurate judgement of the situation by employees is the only reason for the return to an initial (natural) rate of unemployment (i.e. the end of the money illusion, when they finally recognize the actual dynamics of prices and wages).
The other (arbitrary) assumption refers to a special informational asymmetry: whatever employees are unaware of in connection with the changes in (real and nominal) wages and prices can be clearly observed by employers. The new classical version of the Phillips curve was aimed at removing the puzzling additional presumptions, but its mechanism still requires money illusion.