  The IceCube collaboration reported a $\sim 3.5\sigma$ excess of $13\pm5$
neutrino events in the direction of the blazar TXS 0506+56 during a $\sim$6
month period in 2014-2015, as well as the ($\sim3\sigma$) detection of a
high-energy muon neutrino during an electromagnetic flare in 2017. We explore
the possibility that the 2014-2015 neutrino excess and the 2017 multi-messenger
flare are both explained in a common physical framework that relies on the
emergence of a relativistic neutral beam in the blazar jet due to interactions
of accelerated cosmic rays (CRs) with photons. We demonstrate that the neutral
beam model provides an explanation for the 2014-2015 neutrino excess without
violating X-ray and $\gamma$-ray constraints, and also yields results
consistent with the detection of one high-energy neutrino during the 2017
flare. If both neutrino associations with TXS 05065+056 are real, our model
requires that (i) the composition of accelerated CRs is light, with a ratio of
helium nuclei to protons $\gtrsim5$, (ii) a luminous external photon field
($\sim 10^{46}$ erg s$^{-1}$) variable (on year-long timescales) is present,
and (iii) the CR injection luminosity as well as the properties of the
dissipation region (i.e., Lorentz factor, magnetic field, and size) vary on
year-long timescales.
