  For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic
structure reflecting the presence of both additive transfers and multiplicative
norms. The additive structure gives rise to a Mackey functor and the
multiplicative structure yields the additional structure of a Tambara functor.
If $R$ is an $N_\infty$ ring spectrum in the category of genuine $G$-spectra,
then all possible additive transfers are present and $\pi_0 R$ has the
structure of an incomplete Tambara functor. However, if $R$ is an $N_\infty$
ring spectrum in a category of incomplete $G$-spectra, the situation is more
subtle.
  In this paper, we study the algebraic theory of Tambara structures on
incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just
as incomplete Tambara functors have compatibility conditions that control which
systems of norms are possible, bi-incomplete Tambara functors have algebraic
constraints arising from the possible interactions of transfers and norms. We
give a complete description of the possible interactions between the additive
and multiplicative structures.
