The emergence of shell forms in the growth of foraminifera is an essential problem in the morphogenesis of these microorganisms. We present a model of foraminiferal shells that applies a moving-reference system. Previous models have referred to fixed-reference axes and have neglected apertures. Our model focuses on real morphologic characteristics and follows stepwise natural biological processes. It introduces apertures based on minimization of the local communication path and applies three parameters, which are either predetermined or selected at random from given ranges. Expression of stochastic parameters mimics phenotypic variability of a shell. We also present a detailed description of the method with examples of simulated shells and the first step toward analyses of the theoretical morphospace. The morphospace is divided into certain regions (phases) separated by transitional planes (phase transitions). Further prospects for foraminiferal modeling, which should focus on more in-depth models based on realistic intracellular dynamics, are also presented.