  It is well-known that first order phase transitions in the early universe can
be a powerful source of observable stochastic gravitational wave backgrounds.
Any such gravitational wave background must exhibit large-scale anisotropies at
least as large as those seen in the CMB $\sim 10^{-5}$, providing a valuable
new window onto the (inflationary) origins of primordial fluctuations. While
significantly larger fractional anisotropies are possible (for example, in
multi-field inflation) and would be easier to interpret, it has been argued
that these can only be consistent with CMB bounds if the gravitational wave
signal is correspondingly smaller. In this paper, we show that this argument,
which relies on assuming radiation dominance of the very early universe, can be
evaded if there is an era of early matter dominance of a certain robust type.
This allows large gravitational wave anisotropies to be consistent with
observable signals at proposed future gravitational wave detectors. Constraints
from the CMB on large scales, as well as primordial black hole and mini-cluster
formation on small scales, and secondary scalar-induced gravitational waves are
all taken into account.
