Sessile organisms that compete for access to resources by overgrowing each other may risk the local elimination of one sex or the other, as frequently happens within clumps of the dioecious liverwort Marchantia inflexa. A multi-stage, spatially implicit differential-equation model of M. inflexa growing in an isolated patch, analysed in a previous study, indicated that long-term coexistence of the sexes within such patches may be only temporary. Here we derive a spatially explicit, sub-individual-based model to reconsider this interpretation when much more ecological realism is taken into account, including the process of fragmentation. The model tracks temporally discrete growth increments in continuous space, representing growth architecture and the overgrowth process in significant geometric detail. Results remain generally consistent with the absence of long-term coexistence of the sexes in individual patches of Marchantia. Dynamics of sex-specific growth qualitatively resemble those generated by differential-equation models, suggesting that this much simpler framework may be adequate for multi-patch metapopulation models. Direct competition between fragmenting and non-fragmenting clones demonstrates the importance of fragmentation in overgrowth competition. The results emphasize the need for empirical work on mechanisms of overgrowth and for modeling and empirical studies of life history tradeoffs and sex-ratio dynamics in multi-patch systems.