  We use the uncertainty relation between the operators associated to the total
number of particles and to the relative phase of two bosonic modes to construct
entanglement and Einstein-Podolsky-Rosen steering criteria. These can be tested
experimentally in a variety of systems, such as optical fields, Bose-Einstein
condensates or mechanical oscillators. While known entanglement criteria
involving the phase observable typically require to perform interference
measurements by recombining the two systems, our criteria can be tested through
local measurements at two spatially distinct positions, to investigate the
nonlocal nature of quantum correlations. We present simple examples where our
criteria are violated, and show their robustness to noise. Apart from being
useful for state characterization, they might find application in quantum
information protocols, for example based on number-phase teleportation.
