Although your feel painful and eyes watering when choking in your dinner, it is normal for you. As you choke, the throat, nose and eye nerves are interlinked which may have the reaction together. You just need to find a place to sit and drink some water to release the symptom.
I think it is natural to make eyes watering when you choke when you eat food. You know, our eyes will tearing when we feel sad, exciting, moved as well as pain. When you got choke, it is really painful. So, it is naturally to makes tears. It is a normal physical process. So. don't worry. It will be recover soon after you recover from choke. Good luck and don't be choked again.
Part of tear secreted by tear gland is used to moisturize human eyes while the other part flow trough lacrimal canalicular laceration, larcinal sac and nasal tube. These three parts consist the passage of tear. When people get choked, foods or drink will enter the air tube so that the respiratory tract is obstructed. The tear passage and respiratory tract share a same part, nasal tube. When respiratory tract is obstructed, tear passage is obstructed too. In that case, both tear and air can not get through the nasal smoothly. People cough to get the obstruction out and to breath. The tears then just flow the other way out through eyes. That is why eyes will be watery when choked.