Recall that a real C*-algebra is a Banach $\ast$-algebra $A$ over $\mathbb{R}$ which satisfies the standard C* identity and which also has the property that $1 + a^{\ast}a$ is invertible in the unitalization of $A$ for every $a$. This is the "right" definition because the "real Gelfand-Naimark theorem" is true for such algebras: every real C*-algebra is isometrically $\ast$-isomorphic to a norm closed $\ast$-algebra of bounded operators on a real Hilbert space.

Now we turn to von Neumann algebras. A von Neumann algebra is supposed to be a $\ast$-algebra of bounded operators on a (complex) Hilbert space which is closed in the weak topology, or equivalently the strong topology. This can be abstracted to the intrinsic definition of a von Neumann algebra as a C* algebra which is the dual of some (complex) Banach space. My question is: what is the intrinsic definition of a real von Neumann algebra which abstracts the notion of a $\ast$-algebra of bounded operators on a real Hilbert space which is closed in the weak topology or (equivalently?) the strong topology?