In this paper, we prove that a pseudo-exponential field has continuum many nonisomorphic countable real closed exponential subfields, each with an order-preserving exponential map which is surjective onto the nonnegative elements. Indeed, this is true of any algebraically closed exponential field satisfying Schanuel’s conjecture.
"Real Closed Exponential Subfields of Pseudo-Exponential Fields." Notre Dame J. Formal Logic 54 (3-4) 591 - 601, 2013. https://doi.org/10.1215/00294527-2143925