On the coincidence of types of a real $AW^*$-algebra and its complexification

We consider real $AW^*$-algebras, that is, Kaplansky algebras over the field of real numbers. As in the case of complex von Neumann algebras and complex $AW^*$-algebras, real $AW^*$-algebras are classified in terms of types $\mathrm{I}_{\mathrm{fin}}$, $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$. We prove that if the complexification $M=A+iA$ of a real $AW^*$-algebra A also is an $AW^*$-algebra, then the types of $A$ and $M$ coincide.