  It is proved that a commutative algebra $A$ of operators in a reflexive real
Banach space has an invariant subspace if each operator $T\in A$ satisfies the
condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }
\varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This
implies the existence of an invariant subspace for every commutative family of
essentially selfadjoint operators in a real Hilbert space.
