Degradation tags, otherwise known as degrons, are portable sequences that can be used to alter protein stability. Here, we report that degron-tagged proteins compete for cellular degradation resources in engineered mammalian cells leading to coupling of the degradation rates of otherwise independently expressed proteins when constitutively targeted human degrons are adopted. We show the effect of this competition to be dependent on the context of the degrons. By considering different proteins, degron position and cellular hosts, we highlight how the impact of the degron on both degradation strength and resource coupling changes, with identification of orthogonal combinations. By adopting inducible bacterial and plant degrons we also highlight how controlled uncoupling of synthetic construct degradation from the native machinery can be achieved. We then build a genomically integrated capacity monitor tagged with different degrons and confirm resource competition between genomic and transiently expressed DNA constructs. This work expands the characterisation of resource competition in engineered mammalian cells to protein degradation also including integrated systems, providing a framework for the optimisation of heterologous expression systems to advance applications in fundamental and applied biological research.