  In this paper we have established an asymptotically conical Morris-Thorne
wormhole solution supported by anisotropic matter fluid and a global monopole
charge in the framework of a $1+3$ dimensional gravity minimally coupled to a
triplet of scalar fields $\phi^a$, resulting from the breaking of a global
$O(3)$ symmetry. For the anisotropic matter fluid we have considered the
equation of state (EoS) given by $\mathcal{P}_r=\omega \rho$, with a
consequence $\omega<-1$, implying a so-called phantom energy at the throat of
the wormhole which violates the energy conditions. In addition, we study the
weak gravitational lensing effect using the Gauss-Bonnet theorem (GBT) applied
to the wormhole optical geometry. We show that the total deflection angle
consists of a term given by $4\pi^2 \eta^2 $, which is independent from the
impact parameter $b$, and an additional term which depends on the radius of the
wormhole throat $b_0$ as well as the dimensionless constant $\zeta$.
