We review a mean-field analysis and give the details of a correlation function approach for spatially distributed systems subject to multiplicative noise, white in space and time. We confirm the existence of a pure noise-induced reentrant nonequilibrium phase transition in the model introduced in [C. Van den Broeck et al., Phys. Rev. Lett. 73, 3395 (1994)], give an intuitive explanation of its origin, and present extensive simulations in dimension d=2. The observed critical properties are compatible with those of the Ising universality class. © 1997 The American Physical Society.