Summary: We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution \( (B, \tau) \) will be a Galois extension of the fixed field of \( \tau \) and will “real split” \( (B, \tau) \). As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras over number fields, considered by G. Berhuy in the context of unitary space-time coding [Adv. Math. Commun. 8, No. 2, 167–189 (2014; Zbl 1328.94096)], is also necessary, proving that Berhuy’s construction is optimal.