  The thermal Sunyaev-Zeldovich (SZ) effect presents a relatively new tool for
characterizing galaxy cluster merger shocks, traditionally studied through
X-ray observations. Widely regarded as the "textbook example" of a cluster
merger bow shock, the western shock front in the Bullet Cluster (1E0657-56)
represents the ideal test case for such an SZ study. We aim to reconstruct a
parametric model for the shock SZ signal by directly and jointly fitting deep,
high-resolution interferometric data from the Atacama Large
Millimeter/submillimeter Array (ALMA) and Atacama Compact Array (ACA) in
Fourier space. The ALMA+ACA data are primarily sensitive to the electron
pressure difference across the shock front. To estimate the shock Mach number
$M$, this difference can be combined with the value for the upstream electron
pressure derived from an independent Chandra X-ray analysis. In the case of
instantaneous electron-ion temperature equilibration, we find
$M=2.08^{+0.12}_{-0.12}$, in $\approx 2.4\sigma$ tension with the independent
constraint from Chandra, $M_X=2.74\pm0.25$. The assumption of purely adiabatic
electron temperature change across the shock leads to $M=2.53^{+0.33}_{-0.25}$,
in better agreement with the X-ray estimate $M_X=2.57\pm0.23$ derived for the
same heating scenario. We have demonstrated that interferometric observations
of the SZ effect provide constraints on the properties of the shock in the
Bullet Cluster that are highly complementary to X-ray observations. The
combination of X-ray and SZ data yields a powerful probe of the shock
properties, capable of measuring $M$ and addressing the question of
electron-ion equilibration in cluster shocks. Our analysis is however limited
by systematics related to the overall cluster geometry and the complexity of
the post-shock gas distribution. To overcome these limitations, a joint
analysis of SZ and X-ray data is needed.
