We propose and analyze a modified ghost-interference experiment, and show that revealing the particle-nature of a particle passing through a double-slit hides the wave-nature of a spatially separated particle which it is entangled with. We derive a nonlocal duality relation, ${mathcal D}_1^2 + {mathcal V}_2^2 le 1$, which connects the path distinguishability of one particle to the interference visibility of the other. It extends Bohrs principle of complementarity to a nonlocal scenario. We also propose a ghost quantum eraser in which, erasing the which-path information of one particle brings back the interference fringes of the other.
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