Electric field control of magnetic properties offers a broad and promising toolbox for enabling ultra-low power electronics. A key challenge with high technological relevance is to master the interplay between the magnetic anisotropy of a ferromagnet and the exchange coupling to an adjacent antiferromagnet. Here, we demonstrate that magneto-ionic gating can be used to achieve a very stable out-of-plane (OOP) oriented magnetization with strong exchange bias in samples with as-deposited preferred in-plane (IP) magnetization. We show that the perpendicular interfacial anisotropy can be increased by more than a factor 2 in the stack Ta/Pt/PtMn/Co/HfO2 by applying −2.5 V gate voltage over 3 nm HfO2, causing a reorientation of the magnetization from IP to OOP with a strong OOP exchange bias of more than 50 mT. Comparing two thicknesses of PtMn, we identify a notable trade-off: while thicker PtMn yields a significantly larger exchange bias, it also results in a slower response to ionic liquid gating within the accessible gate voltage window. These results pave the way for post-deposition electrical tailoring of magnetic anisotropy and exchange bias in samples requiring significant exchange bias.