Fix any real algebraic extension $$\mathbb K$$ of the field $$\mathbb Q$$ of rationals. Polynomials with coefficients from $$\mathbb K$$ in n variables and in n exponential functions are called exponential polynomials over $${\mathbb K}$$ . We study zero sets in $${\mathbb R}^n$$ of exponential polynomials over $$\mathbb K$$ , which we call exponential-algebraic sets. Complements of all exponential-algebraic sets in $${\mathbb R}^n$$ form a Zariski-type topology on $${\mathbb R}^n$$ . Let $$P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$$ be a polynomial and denote $$\begin{aligned} V:=\{ (x_1, \ldots , x_n) \in {\mathbb R}^n|\> P(x_1, \ldots ,x_n,, e^{x_1}, \ldots ,e^{x_n})=0 \}. \end{aligned}$$ The main result of this paper states that, if the real zero set of a polynomial P is irreducible over $$\mathbb K$$ and the exponential-algebraic set V has codimension 1, then, under Schanuel’s conjecture over the reals, either V is irreducible (with respect to the Zariski topology) or each of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of P. In the case of a single exponential (i.e., when P is independent of $$U_2, \ldots , U_n$$ ) stronger statements are shown which are independent of Schanuel’s conjecture.