  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 $P$-representation. 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.
