  Understanding common envelope (CE) evolution is an outstanding problem in
binary evolution. Although the CE phase is not driven by gravitational-wave
(GW) emission, the in-spiraling binary emits GWs that passively trace the CE
dynamics. Detecting this GW signal would provide direct insight into the
gas-driven physics. Even a non-detection might offer invaluable constraints. We
investigate the prospects of detection of a Galactic CE by LISA. While the
dynamical phase of the CE is likely sufficiently loud for detection, it is
short and thus rare. We focus instead on the self-regulated phase that proceeds
on a thermal timescale. Based on population synthesis calculations and the
(unknown) signal duration in the LISA band, we expect $\sim 0.1-100$ sources in
the Galaxy during the mission duration. We map the GW observable parameter
space of frequency $f_\mathrm{GW}$ and its derivative $\dot f_\mathrm{GW}$
remaining agnostic on the specifics of the inspiral, and find that signals with
$\mathrm{SNR}>10$ are possible if the CE stalls at separations such that
$f_\mathrm{GW}\gtrsim2\times10^{-3}\,\mathrm{Hz}$. We investigate the
possibility of misidentifying the signal with other known sources. If the
second derivative $\ddot f_\mathrm{GW}$ can also be measured, the signal can be
distinguished from other sources using a GW braking-index. Alternatively,
coupling LISA with electromagnetic observations of peculiar red giant stars
and/or infrared and optical transients might allow for the disentangling of a
Galactic CE from other Galactic and extra-galactic GW sources.
