Microtubules are polymers required for chromosome segregation. Their drug-induced hyperstabilization impairs chromosome segregation and is an established anti-cancer therapy. How cells respond to microtubule hyperstabilization, however, is incompletely understood. To study this, we evolved budding yeast cells expressing a microtubule-hyperstabilizing tubulin mutant and isolated adapted strains. Aneuploidy of specific chromosomes carrying the microtubule regulators STU2 and VIK1/KAR3 was the first observable adaptation. In the longer run, aneuploidies were outcompeted by mutations in α- or β-tubulin, partially overlapping with mutations in cancer patients. Thus, compensation of microtubule hyperstabilization follows a restrained and reproducible path where new mutations combine with the original offending mutation on the same carrier. While partly compensatory, several mutations failed to re-establish fully normal microtubule dynamics. Sustained growth relied on the mitotic checkpoint, indicating that extended mitotic timing limits the genomic instability caused by reduced microtubule dynamics. Our results predict a potential vulnerability of cells resistant to microtubule-hyperstabilizing agents.