  Let $(\tau,V_\tau)$ be a spinor representation of $\mathrm{Spin}(n)$ and let
$(\sigma,V_\sigma)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that
occurs in the restriction $\tau_{\mid \mathrm{Spin}(n-1)}$.
  We consider the real hyperbolic space $H^n(\mathbb R)$ as the rank one
homogeneous space $\mathrm{Spin}_0(1,n)/\mathrm{Spin}(n)$ and the spinor bundle
$\Sigma H^n(\mathbb R)$ over $H^n(\mathbb R)$ as the homogeneous bundle
$\mathrm{Spin}_0(1,n)\times_{\mathrm{Spin}(n)} V_\tau$.
  Our aim is to characterize eigenspinors of the algebra of invariant
differential operators acting on $\Sigma H^n(\mathbb R)$ which can be written
as the Poisson transform of $L^p$-sections of the bundle
$\mathrm{Spin}(n)\times_{\mathrm{Spin}(n-1)} V_\sigma$ over the boundary
$S^{n-1}\simeq \mathrm{Spin}(n)/\mathrm{Spin}(n-1)$ of $H^n(\mathbb R)$, for
$1<p<\infty$.
