This paper explores the design of reactive distillation columns with vapor/liquid/liquid equilibria. The number of phases and the phase equilibria on each tray are determined by minimizing the Gibbs free energy, allowing for a variable number of phases on each tray. The minimization problems are embedded within a larger problem that minimizes the annualized cost of the column, creating a bilevel optimization problem. A novel sequential approximation approach that can identify global solutions of the inner optimization problems is also introduced. The sequential approximation approach is employed to design reactive distillation and reactive flash processes.