Domain formation in a [Pt/Co] multilayer with out-of-plane anisotropy leads to a shift of the hysteresis loop of an adjacent NiFe thin film with in-plane anisotropy in an analogous manner to the exchange-bias phenomenon typically observed in ferromagnetic/antiferromagnetic systems. This paper shows that the loop shift can be gradually tuned through modification of the magnitude of the perpendicular anisotropy of the [Pt/Co]/NiFe system by simply varying the thickness of an underlying Pt buffer layer. Magnetometry, magnetotransport, and magnetic force microscopy experimental measurements were made. It appears that the key parameter which controls the exchange-bias effect in this system is the presence of a vortex core in between magnetic domains in the [Pt/Co] multilayer. The magnitude of the exchange-bias is related to the ability to unpin this vortex core, which is related to the magnitude of the perpendicular anisotropy. Micromagnetic simulations illustrate the magnetic configuration responsible for the exchange-bias phenomenon in this nonstandard exchange-biased type system.