Conditional probabilities for a single photon at a beam splitter

Eight-port homodyne detection may be used to perform a joint measurement on a single-mode input to a beam splitter; the $x$ quadrature is measured at one output port, and the $p$ quadrature is measured at the other. This is possible even when the input mode contains only one photon. We show to what extent this "joint measurement of a single photon" may be reconciled with the semiclassical notion that the photon may only exit from one of the beam splitter output ports.