  The non-trivial third homotopy class of three-dimensional topological
insulators leads to quantized, magneto-electric coefficient or axion angle
$\theta= n \pi$, with $n \in \mathbb{Z}$. In Part I, we developed tools for
computing $n$ from a staggered symmetry-indicator $\kappa_{AF,j}$ and Wilson
loops of non-Abelian, Berry connection in momentum-space, which clearly
distinguished between magneto-electrically trivial ($n=0$), and non-trivial
($n=2s$) topological crystalline insulators. In this work, we perform
$\mathbb{Z}$-classification of real-space, topological response or $\theta$ by
carrying out thought experiments with magnetic, Dirac monopoles. We demonstrate
this for non-magnetic and magnetic topological insulators by computing induced
electric charge on monopoles or Witten effect. We show that both first- and
higher- order topological insulators can exhibit quantized, magneto-electric
response, irrespective of the presence of gapless surface-states, and
corner-states. Special attention is paid to the response of octupolar
higher-order topological insulator, which was originally predicted to be
magneto-electrically trivial. The important roles of fermion zero-modes,
$\mathcal{CP}$, and flavor symmetries are critically addressed. Our work
outlines a unified theoretical framework for addressing dc topological response
and topological quantum phase transitions, which cannot be reliably predicted
by symmetry-based classification scheme.
