  We consider the problem of computing satisfactory pairs of solutions of the
differential equation for Legendre functions of non-negative integer order
$\mu$ and degree $-\frac12+i\tau$, where $\tau$ is a non-negative real
parameter. Solutions of this equation are the conical functions
${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ and ${Q}^{\mu}_{-\frac12+i\tau}(x)$,
$x>-1$. An algorithm for computing a numerically satisfactory pair of solutions
is already available when $-1<x<1$ (see \cite{gil:2009:con},
\cite{gil:2012:cpc}).In this paper, we present a stable computational scheme
for a real valued numerically satisfactory companion of the function
${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ for $x>1$, the function
$\Re\left\{e^{-i\pi \mu} {{Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The
proposed algorithm allows the computation of the function on a large parameter
domain without requiring the use of extended precision arithmetic.
