  Primordial fluctuations in the relative number densities of particles, or
isocurvature perturbations, are generally well constrained by cosmic microwave
background (CMB) data. A less probed mode is the compensated isocurvature
perturbation (CIP), a fluctuation in the relative number densities of cold dark
matter and baryons. In the curvaton model, a subdominant field during inflation
later sets the primordial curvature fluctuation $\zeta$. In some curvaton-decay
scenarios, the baryon and cold dark matter isocurvature fluctuations nearly
cancel, leaving a large CIP correlated with $\zeta$. This correlation can be
used to probe these CIPs more sensitively than the uncorrelated CIPs considered
in past work, essentially by measuring the squeezed bispectrum of the CMB for
triangles whose shortest side is limited by the sound horizon. Here, the
sensitivity of existing and future CMB experiments to correlated CIPs is
assessed, with an eye towards testing specific curvaton-decay scenarios. The
planned CMB Stage 4 experiment could detect the largest CIPs attainable in
curvaton scenarios with more than 3$\sigma$ significance. The significance
could improve if small-scale CMB polarization foregrounds can be effectively
subtracted. As a result, future CMB observations could discriminate between
some curvaton-decay scenarios in which baryon number and dark matter are
produced during different epochs relative to curvaton decay. Independent of the
specific motivation for the origin of a correlated CIP perturbation,
cross-correlation of CIP reconstructions with the primary CMB can improve the
signal-to-noise ratio of a CIP detection. For fully correlated CIPs the
improvement is a factor of $\sim$2$-$3.
