  The recent detection of possible neutrino emission from the blazar TXS
0506+056 was the first high-energy neutrino associated with an astrophysical
source, making this special type of active galaxies promising neutrino
emitters. The fact that two distinct episodes of neutrino emission were
detected with a separation of around 3 years suggests that emission could be
periodic. Periodic emission is expected from supermassive binary black hole
systems due to jet precession close to the binary's merger. Here we show that
if TXS 0506+056 is a binary source then the next neutrino flare could occur
before the end of 2021. We derive the binary properties that would lead to the
detection of gravitational waves from this system by LISA. Our results for the
first time quantify the time scale of these correlations for the example of TXS
0506+056, providing clear predictions for both the neutrino and
gravitational-wave signatures of such sources.
