  Baker and Bowler defined a category of algebraic objects called tracts which
generalize both partial fields and hyperfields. They also defined a notion of
weak and strong matroids over a tract $F$, and proved that if $F$ is perfect,
meaning that $F$-vectors and $F$-covectors are orthogonal for every matroid
over $F$, then the notions of weak and strong $F$-matroids coincide. We define
the class of strongly fused tracts and prove that such tracts are perfect. We
in fact prove a more general result which implies that given a tract $F$, there
is a tract $\sigma (F)$ with the same 3-term additive relations as $F$ such
that weak $F$-matroids coincide with strong $\sigma (F)$-matroids. We also show
that both partial fields and stringent hyperfields are strongly fused; in this
way, our criterion for perfection generalizes results of Baker-Bowler and
Bowler-Pendavingh.
