A Monte Carlo algorithm is defined for generating replicas of textile composite specimens that possess the same statistical characteristics as specimens imaged using high resolution computed tomography. The textile reinforcement is represented by one-dimensional tow loci in three-dimensional space, which are easily incorporated into the Binary Model of textile composites. A tow locus is expressed as the sum of non-stochastic, periodic variations in the coordinates of the tow centroid and stochastic, non-periodic deviations. The non-stochastic variations have period commensurate with the dimensions of the unit cell of the textile, while the stochastic deviations, which describe geometrical defects, exhibit correlation lengths that may be incommensurate with the unit cell. The model is calibrated with data deduced in prior work from computed tomography images. The calibration obviates the need for assuming any ideal shape functions for the tow loci, which can take very general form. The approach is therefore valid for a wide range of textile architectures. Once calibrated, a Markov Chain algorithm can generate numerous stochastic replicas of a textile architecture very rapidly. These virtual specimens can be much larger than the real specimens from which the data were originally gathered, a necessary feature when real specimen size is limited by the nature of high resolution computed tomography. The virtual specimen generator is illustrated using data for an angle interlock weave.