  A conservative constraint on the Einstein Weak Equivalence Principle (WEP)
can be obtained under the assumption that the observed time delay between
correlated particles from astronomical sources is dominated by the
gravitational fields through which they move. Current limits on the WEP are
mainly based on the observed time delays of photons with different energies. It
is highly desirable to develop more accurate tests that include the
gravitational wave (GW) sector. The detection by the advanced LIGO/VIRGO
systems of gravitational waves will provide attractive candidates for
constraining the WEP, extending the tests to gravitational interactions, with
potentially higher accuracy. Considering the capabilities of the advanced
LIGO/VIRGO network and the source direction uncertainty, we show that the joint
detection of GWs and electromagnetic signals could probe the WEP to an accuracy
down to $10^{-10}$, which is one order of magnitude tighter than previous
limits, and seven orders of magnitude tighter than the multi-messenger (photons
and neutrinos) results by supernova 1987A.
