  In this paper we consider the diffusive search for a bounded target $\Omega
\in \R^d$ with its boundary $\partial \Omega$ totally absorbing. We assume that
the target is surrounded by a semipermeable interface given by the closed
surface $\partial \calM$ with $\Omega \subset \calM\subset \R^d$. That is, the
interface totally surrounds the target and thus partially screens the diffusive
search process. We also assume that the position of the diffusing particle
(searcher) randomly resets to its initial position $\x_0$ according to a
Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be
outside the interface, $\x_0\in \calM^c$, which means that resetting does not
occur when the particle is within the interior of $\partial \calM$. Hence, the
semipermeable interface also screens out the effects of resetting. We first
solve the boundary value problem (BVP) for diffusion on the half-line $x\in
[0,\infty)$ with an absorbing boundary at $x=0$, a semipermeable barrier at
$x=L$, and stochastic resetting to $x_0>L$ for all $x>L$. We calculate the mean
first passage time (MFPT) to be absorbed by the target and explore its behavior
as a function of the permeability $\kappa_0$ of the interface and its spatial
position $L$. We then perform the analogous calculations for a
three-dimensional (3D) spherically symmetric interface and target, and show
that the MFPT exhibits the same qualitative behavior as the 1D case. Finally,
we introduce a stochastic single-particle realization of the search process
based on a generalization of so-called snapping out BM. The latter sews
together successive rounds of reflecting Brownian motion on either side of the
interface. The main challenge is establishing that the probability density
generated by the snapping out BM satisfies the permeable boundary conditions at
the interface. We show how this can be achieved using renewal theory.
