  We investigate valuated matroids with an additional algebraic structure on
their residue matroids. We encode the structure in terms of representability
over stringent hyperfields.
  A hyperfield $H$ is {\em stringent} if $a\boxplus b$ is a singleton unless
$a=-b$, for all $a,b\in H$. By a construction of Marc Krasner, each valued
field gives rise to a stringent hyperfield.
  We show that if $H$ is a stringent skew hyperfield, then the vectors of any
weak matroid over $H$ are orthogonal to its covectors, and we deduce that weak
matroids over $H$ are strong matroids over $H$. Also, we present vector axioms
for matroids over stringent skew hyperfields which generalize the vector axioms
for oriented matroids and valuated matroids.
