  We consider the action of a special class of reciprocal transformation on the
principal hierarchy associated to a semisimple $F$-manifold with compatible
flat structure $(M,\circ,\nabla,e)$. Under some additional assumptions, the
hierarchy obtained applying these reciprocal transformations is also associated
to an $F$-manifold with compatible flat structure that we call reciprocal
$F$-manifold. We also consider the special case of bi-flat $F$-manifolds
$(M,\circ,*,\nabla^{(1)},\nabla^{(2)},e,E)$ and we study reciprocal
transformations preserving flatness of both the connections $\nabla^{(1)}$ and
$\nabla^{(2)}$ and how they act on corresponding solutions of an augmented
Darboux-Egorov system.
