Abstract: Guarded existential rules form a robust rule-based language for modelling ontologies. The central problem of ontology-based query answering, as well as the notion of polynomial combined rewritability, have been extensively studied during the last years for this formalism. However, the relevant setting where the underlying signature is considered to be fixed is far from being well understood. All the existing results on ontology-based query answering and polynomial combined rewritability assume rule heads with one atom, while existential rules in real ontologies are typically coming with multi-heads consisting of several atoms. We aim to fill this gap.