  One can associate to a valued field an inverse system of valued hyperfields
$(\mathcal{H}_i)_{i \in I}$ in a natural way. We investigate when, conversely,
such a system arise from a valued field. First, we extend a result of Krasner
by showing that the inverse limit of certain systems are stringent valued
hyperfields. Secondly, we describe a Hahn-like construction which yields a
henselian valued field from a stringent valued hyperfield. In addition, we
provide an axiomatisation of the theory of stringent valued hyperfields in a
language consisting of two binary function symbols $\oplus$ and $\cdot$ and two
constant symbols $\textbf{0}$ and $\textbf{1}$.
