  In this paper, we study the classification of Lipschitz global solutions for
a two-phase $p$-Laplace Bernoulli problem, subject to a mild assumption.
Specifically, we focus on the scenario where the interior two-phase points of
the global solution are non-empty. Our results show that the expected
$C^{1,\eta}$ regularity holds in a suitable neighborhood of certain two-phase
points referred to as regular two-phase points.
