  We prove necessary and sufficient conditions for the informational
completeness of an arbitrary set of Gaussian observables on continuous variable
systems with finite number of degrees of freedom. In particular, we show that
an informationally complete set either contains a single informationally
complete observable, or includes infinitely many observables. We show that for
a single informationally complete observable, the minimal outcome space is the
phase space, and the observable can always be obtained from the quantum optical
$Q$-function by linear postprocessing and Gaussian convolution, in a suitable
symplectic coordinatization of the phase space. In the case of projection
valued Gaussian observables, e.g., generalized field quadratures, we show that
an informationally complete set of observables is necessarily infinite.
Finally, we generalize the treatment to the case where the measurement coupling
is given by a general linear bosonic channel, and characterize informational
completeness for an arbitrary set of the associated observables.
