  The evolution of cosmic structures, the formation and growth of the first
black holes and the connection to their baryonic environment are key unsolved
problems in astrophysics. The X-ray Athena mission and the gravitational-wave
Laser Interferometer Space Antenna (LISA) offer independent and complementary
angles on these problems. We show that up to 10 black hole binaries in the mass
range 10^5 - 10^8 Msun discovered by LISA at redshift <~ 3.5 could be detected
by Athena in an exposure time up to 100 ks, if prompt X-ray emission of ~ 1% -
10% of the Eddington luminosity is present. Likewise, if any LISA-detected
extreme mass ratio inspirals occur in accretion disks, Athena can detect
associated electromagnetic emission out to redshift ~ 1. Finally, warned by
LISA, Athena can point in advance and stare at stellar-mass binary black hole
mergers at redshift <~ 0.1. These science opportunities emphasise the vast
discovery space of simultaneous observations from the two observatories, which
would be missed if they were operated in different epochs.
