  Ferroics, characterized by a broken symmetry state with nonzero elastic,
polar, or magnetic order parameters $\vec{u}$, are recognized platforms for
staging and manipulating topologically-protected structures as well as for
detecting unconventional topological phenomena. The unrealized possibility of
producing ordered topological phases in magnetoelectric multiferroics,
exhibiting coupled magnetic and polar order parameters, is anticipated to
engender novel functionality and open avenues for manipulating topological
features. Here, we report the discovery of an ordered $\pi_1$-$S_\infty$ vortex
phase within single-phase magnetoelectric multiferroic BiFeO$_3$. The phase,
characterized by positive topological charge and chiral staggering, is realized
in coherent TbScO$_3$ and BiFeO$_3$ superlattices and established via the
combination of direct- and Fourier-space analyses. Observed order-parameter
morphologies are reproduced with a field model describing the local
order-parameter stiffness and competing non-local dipole-dipole interactions.
Anisotropies canting the order parameter towards $\left<100\right>$ suppress
chiral staggering and produced a competing $\pi_1$-$C_{\infty v}$ vortex phase
in which cores are centered.
