Bistable switches are fundamental regulatory elements of complex systems, ranging from electronics to living cells. Designed genetic toggle switches have been constructed from pairs of natural transcriptional repressors wired to inhibit one another. The complexity of the engineered regulatory circuits can be increased using orthogonal transcriptional regulators based on designed DNA-binding domains. However, a mutual repressor-based toggle switch comprising DNA-binding domains of transcription-activator-like effectors (TALEs) did not support bistability in mammalian cells. Here, the challenge of engineering a bistable switch based on monomeric DNA-binding domains is solved via the introduction of a positive feedback loop composed of activators based on the same TALE domains as their opposing repressors and competition for the same DNA operator site. This design introduces nonlinearity and results in epigenetic bistability. This principle could be used to employ other monomeric DNA-binding domains such as CRISPR for applications ranging from reprogramming cells to building digital biological memory.