Holonomic phases--geometric and topological--have long been an intriguing aspect of physics. They are ubiquitous, ranging from observations in particle physics to applications in fault tolerant quantum computing. However, their exploration in particles sharing genuine quantum correlations lacks in observations. Here, we experimentally demonstrate the holonomic phase of two entangled photons evolving locally, which, nevertheless, gives rise to an entanglement-dependent phase. We observe its transition from geometric to topological as the entanglement between the particles is tuned from zero to maximal, and find this phase to behave more resiliently to evolution changes with increasing entanglement. Furthermore, we theoretically show that holonomic phases can directly quantify the amount of quantum correlations between the two particles. Our results open up a new avenue for observations of holonomic phenomena in multiparticle entangled quantum systems.