  Multi-planetary systems are prevalent in our Galaxy. The long-term stability
of such systems may be disrupted if a distant inclined companion excites the
eccentricity and inclination of the inner planets via the eccentric Kozai-Lidov
mechanism. However, the star-planet and the planet-planet interactions can help
stabilize the system. In this work, we extend the previous stability criterion
that only considered the companion-planet and planet-planet interactions by
also accounting for short-range forces or effects, specifically, relativistic
precession induced by the host star. A general analytical stability criterion
is developed for planetary systems with $N$ inner planets and a relatively
distant inclined perturber by comparing precession rates of relevant dynamical
effects. Furthermore, we demonstrate as examples that in systems with $2$ and
$3$ inner planets, the analytical criterion is consistent with numerical
simulations using a combination of Gauss's averaging method and direct N-body
integration. Finally, the criterion is applied to observed systems,
constraining the orbital parameter space of a possible undiscovered companion.
This new stability criterion extends the parameter space in which an inclined
companion of multi-planet systems can inhabit.
