Cross posted from here after no responses and a bounty being placed on the question.

Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-Hirzebruch spectral sequence:

$E_2^{p,q}:=H^p(X;h^q(\ast))\Rightarrow h^{p+q}(X)$.

In the case of complex topological $K$-theory, i.e $KU^n(X)$, the differentials admit nice descriptions in terms of higher cohomology operations (i.e. $d^3=Sq^3$, the Steenrod square of degree $3$). For twisted $KU$ we have that $d^3(-)=Sq^3(-)+ \lambda\smile (-)$ where $H\in H^3(X;\mathbb{Z})$ is the class of the twist.

This may be a naive question but is there a similar description for real topological (twisted) $K$-theory, $KO$? I am particularly interested in $d^3$.