  Estimation of the phase delay between interferometer arms is the core of
transmission phase microscopy. Such phase estimation may exhibit an error below
the standard quantum (shot-noise) limit, if the input is an entangled two-mode
state, e.g., a N00N state. We show, by contrast, that such supersensitive phase
estimation (SSPE) is achievable by \textit{incoherent}, e.g., \textit{thermal},
light that is injected into a Mach-Zehnder interferometer via a Kerr-nonlinear
two-mode coupler. Phase error is shown to be reduced below $1/\bar{n}$,
$\bar{n}$ being the mean photon number, by thermal input in such
interferometric setups, even for small nonlinear phase-shifts per photon pair
or for significant photon loss. Remarkably, the phase accuracy achievable in
such setups by thermal input surpasses that of coherent light with the same
$\bar{n}$. Available mode couplers with giant Kerr nonlinearity that stems
either from dipole-dipole interactions of Rydberg polaritons in a cold atomic
gas, or from cavity-enhanced dispersive atom-field interactions, may exploit
such effects to substantially advance interferometric phase microscopy using
incoherent, faint light sources.
