  A vector field E on an F-manifold (M, o, e) is an eventual identity if it is
invertible and the multiplication X*Y := X o Y o E^{-1} defines a new
F-manifold structure on M. We give a characterization of such eventual
identities, this being a problem raised by Manin. We develop a duality between
F-manifolds with eventual identities and we show that is compatible with the
local irreducible decomposition of F-manifolds and preserves the class of
Riemannian F-manifolds. We find necessary and sufficient conditions on the
eventual identity which insure that harmonic Higgs bundles and DChk-structures
are preserved by our duality. We use eventual identities to construct
compatible pair of metrics.
