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 stringent if a⊞b is a singleton unless a=−b, for all a,b∈H
Nearby in the stack
. 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.