Using analog values with Algebraic Normal Form?
Perhaps the most appropriate way of thinking of algebraic normal form is as the following statement:


  Every function from $\mathbb{Z}_2^n$ to $\mathbb{Z}_2$ can be represented uniquely as a multilinear polynomial.


Here $\mathbb{Z}_2$ is the field with two elements, and a polynomial is multilinear if all monomials are products of distinct variables (so you can't have $x_i^2$ in a monomial).

The upshot is that the modulo 2 operation is not the most natural way of understanding the computation involved in algebraic normal form; rather, addition is done over the field $\mathbb{Z}_2$. It seems rather unnatural to apply the modulo 2 operation to real numbers in this context.

One generalization of algebraic normal form to real numbers is the Fourier expansion over the Boolean cube:


  Every function from $\{-1,1\}^n$ to $\mathbb{R}$ can be represented uniquely as a multilinear polynomial (with real coefficients).


Here $\mathbb{R}$ is the field of real numbers.

Another generalization is the Hermite expansion in Gaussian space:


  Every function from $\mathbb{R}^n$ to $\mathbb{R}$ can be approximated arbitrarily well by a polynomial (with real coefficients).


Here arbitrarily well means up to $\epsilon$ distance in $L^2$ with respect to Gaussian measure (e.g., see this Wikipedia article).