Coreference Resolution (CR) is a crucial yet challenging task in natural language understanding, often constrained by task-specific architectures and encoder-based language models that demand extensive training and lack adaptability. This study introduces the first multilingual CR methodology which leverages decoder-only LLMs to handle both overt and zero mentions. The article explores how to model the CR task for LLMs via five different instruction sets using a controlled inference method. The approach is evaluated across three LLMs; Llama 3.1, Gemma 2, and Mistral 0.3. The results indicate that LLMs, when instruction-tuned with a suitable instruction set, can surpass state-of-the-art task-specific architectures. Specifically, our best model, a fully fine-tuned Llama 3.1 for multilingual CR, outperforms the leading multilingual CR model (i.e., Corpipe 24 single stage variant) by 2 pp on average across all languages in the CorefUD v1.2 dataset collection.