In what sense are fields an algebraic theory?

Fields are not algebraic. An algebraic theory, for example, has free objects: there are free rings, free groups, a free monoids. The free functor is left adjoint to the forgetful functor to sets (okay, I'm talking about models in Set). There are, though, no free fields.

One can extend one's idea of an "algebraic theory" to an "essentially algebraic theory" in which partially defined operations are allowed (it's not clear to me that fields satisfy those since you need to specify the domain in terms of other operations whereas it seems that one can only specify the domain of the inverse as the complement of such a subset). Or, (maybe, but I doubt it), one could define a field as a Z₂-graded algebraic theory where 0 is in degree 0 and everything else is in degree 1. Here, a grading should be regarded simply as a labelling system.

Alternatively, one can talk about meadows. Meadows are algebraic theories which are modified versions of fields. Instead of multiplicative inverses, there is a unary operation ι:M → M which satisfies the identity xι(x)x = x. Defining ι(x) = x^-1 for non-zero x, and ι(0) = 0 turns any field into a meadow. The relationship between meadows and fields is quite strong.

An arXiv search throws up 68 references (at time of writing; for some reason google doesn't turn up anything particularly relevant, even when combined with the word "field"). One prominent name is that of Jan Bergstra.