  Given an $F$-manifold with eventual identities we examine what this structure
entails from the point of view of integrable PDEs of hydrodynamic type. In
particular, we show that in the semisimple case the characterization of
eventual identities recently given by David and Strachan is equivalent to the
requirement that $E\circ$ has vanishing Nijenhuis torsion. Moreover, after
having defined new equivalence relations for connections compatible with
respect to the $F$-product $\circ$, namely hydrodynamically almost equivalent
and hydrodynamically equivalent connections, we show how these two concepts
manifest themselves in several specific situations. In particular, in the case
of an $F$-manifold endowed with eventual identity and two almost
hydrodynamically equivalent flat connections we are able to derive the
recurrence relations for the flows of the associated integrable hierarchy. If
the two connections originate from a flat pencil of metrics these reduce to the
standard bi-Hamiltonian recursion. Furthermore, using the geometric set-up
proposed here we show how the recurrence relations of the principal hierarchy
introduced by Dubrovin arise in this general framework and we provide a general
cohomological set-up for the conservation laws of the semihamiltonian hierarchy
associated to a semisimple $F$-manifold with compatible connection and eventual
identity. Therefore, the point of view we propose, not only highlight the
conceptual unity of two well-known recursive schemes (principal hierarchy and
classical bi-Hamiltonian) but it also provides a far reaching generalization of
these recursions that relies on the presence of an eventual identity.
