  The ability of matter to be superposed at two different locations while being
intrinsically connected by a quantum phase is among the most counterintuitive
predictions of quantum physics. While such superpositions have been created for
a variety of systems, the in-situ observation of the phase coherence has
remained out of reach. Using a heralding measurement on a spin-oscillator
entangled state, we project a mechanical trapped-ion oscillator into a
superposition of two spatially separated states, a situation analogous to
Schr\"odinger's cat. Quantum interference is clearly observed by extracting the
occupations of the energy levels. For larger states, we encounter problems in
measuring the energy distribution, which we overcome by performing the
analogous measurement in a squeezed Fock basis with each basis element
stretched along the separation axis. Using 8 dB of squeezing we observe quantum
interference for cat states with phase space separations of $\Delta \alpha =
15.6$, corresponding to wavepackets with a root-mean-square extent of 7.8 nm
separated by over 240 nm. We also introduce a method for reconstructing the
Wigner phase-space quasi-probability distribution using both squeezed and
non-squeezed Fock bases. We apply this to a range of negative parity cats,
observing the expected interference fringes and negative values at the center
of phase space. Alongside the fundamental nature of these large state
superpositions, our reconstruction methods facilitate access to the large
Hilbert spaces required to work with mesoscopic quantum superpositions, and may
be realized in a wide range of experimental platforms.
