In a market economy with inflationary fluctuations and stochastic rate of return, we study robust Nash equilibrium strategies for two competing asset-liability fund managers. The two fund managers are averse to model uncertainty and prone to money illusion, which leads them to partially overlook inflation risks. The financial market consists of a nominal bond, an inflation-linked index bond, and a stock. The inflation index is characterized using a continuous-time Fisher equation, while the expected rate of return on the stock is modeled by a mean-reverting Ornstein-Uhlenbeck process, which captures the market dynamics of bull and bear phases. Each fund manager aims to develop a robust investment strategy to outperform their competitors by maximizing the expected terminal utility of a composite measure of relative nominal and real surpluses under worst-case scenarios. The robust two-player stochastic differential game is addressed using a backward stochastic differential equation approach, and explicit expressions of robust Nash equilibrium investment strategies, density generator processes under well-defined worst-case misspecification, and the corresponding value functions are derived. The financial implications of money illusion, model ambiguity, and strategic competition are analyzed, and several particular results are recovered. Finally, the economic effects of some model parameters on robust investment policies are demonstrated with numerical examples.