  Joint observations of gravitational waves and electromagnetic counterparts
will answer questions about cosmology, gamma-ray bursts, and the behaviour of
matter at supranuclear densities. The addition of a Southern-Hemisphere
gravitational-wave observatory to proposed global networks creates a longer
baseline, which is beneficial for sky localisation. We analyse how an
observatory in Australia can enhance the multi-messenger astronomy capabilities
of future networks. We estimate the number of binary neutron star mergers with
joint observations of gravitational waves and kilonova counterparts detectable
by the Vera C. Rubin Observatory. First, we consider a network of upgrades to
current observatories. Adding an Australian observatory to a three-observatory
network (comprising two observatories in the USA and one in Europe) boosts the
rate of joint observations from $2.5^{+4.5}_{-2.0}$ per year to
$5.6^{+10}_{-4.5}$ per year (a factor of two improvement). Then, we consider a
network of next-generation observatories. Adding a $20$ km Australian
observatory to a global network of a Cosmic Explorer $40$ km in the USA and an
Einstein Telescope in Europe only marginally increases the rate from
$40^{+71}_{-32}$ per year to $44^{+79}_{-35}$ per year (a factor of 1.1
improvement). The addition of an Australian observatory, however, ensures that
at least two observatories are online far more often. When the Cosmic Explorer
$40$ km is offline for a major upgrade, the Australian observatory increases
the joint observation rate from $0.5^{+0.8}_{-0.4}$ per year to
$38^{+68}_{-30}$ per year (a factor of 82 improvement). When the Einstein
Telescope is offline, the joint observation rate increases from
$0.2^{+0.3}_{-0.1}$ per year to $19^{+34}_{-15}$ per year (a factor of 113
improvement). We sketch out the broader science case for a Southern-Hemisphere
gravitational-wave observatory.
