Cross posted to MathOverflow after no response with a bounty.
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 $H$ is the class of the twist.
Is there a similar description for real topological (twisted) $K$-theory $KO$? I am particularly interested in $d^3$.