Chromosome and genome stability are important for normal cell function as instability often correlates with disease and dysfunction of DNA repair mechanisms. Many organisms maintain supernumerary or accessory chromosomes that deviate from standard chromosomes. The pathogenic fungus Zymoseptoria tritici has as many as eight accessory chromosomes, which are highly unstable during meiosis and mitosis, transcriptionally repressed, show enrichment of repetitive elements, and enrichment with heterochromatic histone methylation marks, e.g., trimethylation of H3 lysine 9 or lysine 27 (H3K9me3, H3K27me3). To elucidate the role of heterochromatin on genome stability in Z. tritici, we deleted the genes encoding the methyltransferases responsible for H3K9me3 and H3K27me3, kmt1 and kmt6, respectively, and generated a double mutant. We combined experimental evolution and genomic analyses to determine the impact of these deletions on chromosome and genome stability, both in vitro and in planta. We used whole genome sequencing, ChIP-seq, and RNA-seq to compare changes in genome and chromatin structure, and differences in gene expression between mutant and wildtype strains. Analyses of genome and ChIP-seq data in H3K9me3-deficient strains revealed dramatic chromatin reorganization, where H3K27me3 is mostly relocalized into regions that are enriched with H3K9me3 in wild type. Many genome rearrangements and formation of new chromosomes were found in the absence of H3K9me3, accompanied by activation of transposable elements. In stark contrast, loss of H3K27me3 actually increased the stability of accessory chromosomes under normal growth conditions in vitro, even without large scale changes in gene activity. We conclude that H3K9me3 is important for the maintenance of genome stability because it disallows H3K27me3 in regions considered constitutive heterochromatin. In this system, H3K27me3 reduces the overall stability of accessory chromosomes, generating a "metastable" state for these quasi-essential regions of the genome.