Multiple photon subtraction applied to a displaced
phase-averaged coherent state, which is a non-Gaussian classical state, produces conditional states
with a non trivial (positive) Glauber-Sudarshan Prepresentation.
We theoretically and experimentally
demonstrate that, despite its simplicity, this class of conditional
states cannot be fully characterized by direct detection
of photon numbers. In particular, the non-Gaussianity
of the state is a characteristics that must be assessed by
phase-sensitive measurements. We also show that the
non-Gaussianity of conditional states can be manipulated
by choosing suitable conditioning values and composition
of phase-averaged states.