Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities.
This doesn't make sense over $\mathbb{C}$ because it isn't an ordered field, but we do have something not present over $\mathbb{R}$ : a nontrivial automorphism, namely complex conjugation.
So, instead of replacing polynomial equations with polynomial inequalities we can instead replace those polynomials in $z_i$ with polynomial in $z_i$ and $\bar{z_i}$
For example, we could have the equation $z = \bar{z}$, which defines the real line.
Is there a name for this theory, the study of graphs of polynomials in complex variables and their conjugates? Note, I am not looking for Hodge theory, which starts with a complex variety and then looks at differential forms in conjugate variables -- I want to start with a generalised variety that uses conjugate variables in its definition, as in the above examples of the real line.