  For a "genuine" equivariant commutative ring spectrum $R$, $\pi_0(R)$ admits
a rich algebraic structure known as a Tambara functor. This algebraic structure
mirrors the structure on $R$ arising from the existence of multiplicative norm
maps. Motivated by the surprising fact that Bousfield localization can destroy
some of the norm maps, in previous work we studied equivariant commutative ring
structures parametrized by $N_\infty$ operads. In a precise sense, these
interpolate between "naive" and "genuine" equivariant ring structures.
  In this paper, we describe the algebraic analogue of $N_\infty$ ring
structures. We introduce and study categories of incomplete Tambara functors,
described in terms of certain categories of bispans. Incomplete Tambara
functors arise as $\pi_0$ of $N_\infty$ algebras, and interpolate between Green
functors and Tambara functors. We classify all incomplete Tambara functors in
terms of a basic structural result about polynomial functors. This
classification gives a conceptual justification for our prior description of
$N_\infty$ operads and also allows us to easily describe the properties of the
category of incomplete Tambara functors.
