A tube of radius $r\;(0<r<\infty)$ around totally real totally geodesic 
$\mathbb{R}H^n(c/4)$ is called a homogeneous real hypersurface of type (B) 
in $\mathbb{C}H^n(c)$. It is known that every type (B) hypersurface 
with radius $r\not=(1/\sqrt{|c|\,})\log_e(2+\sqrt{3\,})$ 
has three distinct constant principal curvatures and 
in the case of $r=(1/\sqrt{|c|\,})\log_e(2+\sqrt{3\,})$ 
the real hypersurface $M^{2n-1}$ of type (B) has 
two distinct constant principal curvatures.
The main purpose of this paper is to characterize this real hypersurface $M$.