  We analyze Bell inequalities violations in photonic experiments for which the
measurement apparatuses are restricted to homodyne measurements. Through
numerical optimization of the Clauser-Horne-Shimony-Holt inequality over
homodyne measurements and binning choices, we demonstrate large violations for
states with a bounded number of photons. When considering states defined within
qubit local subspaces of two Fock states, such as NOON states, a violation is
observed solely within the qubit Fock space spanned by zero and two photons.
For more generic states, large violations are obtained. Significant violations
are observed even for states containing three photons locally and under
realistic values of noise and losses. We propose concrete implementations to
achieve such violations, opening new avenues for Bell experiments with homodyne
detectors.
