The concept of overall or effective degree of coherence has proven useful and insightful in the scalar theory of light. In this paper we extend the concept to fields represented over arbitrary vector (Hilbert) spaces. We show that the effective degree of coherence remains unaltered by scaled unitary mappings, which describe for instance many physical interactions. In the context of random, vector-valued electromagnetic fields the invariance implies that the effective degree of coherence assumes the same value in commonly used representation spaces. Thereby, in contrast to the local degree of coherence, this quantity can be taken as an intrinsic property of the electromagnetic wave field. In particular, the effective degree of coherence does not change when the field is scattered by a lossless medium and it can be determined from the far-field pattern. Hence the effective degree of coherence of electromagnetic fields is readily measurable.