Story of the play. Romeo and Juliet is set in Verona, where a feud between the Montague and Capulet families often leads to violence. (Prologue) The Chorus tells, briefly, the story of the play. Romeo and Juliet, the story. Arthur Brooke, The Tragicall Historye of Romeus and Iuliet, 1562.
From there, they drain into the nasolacrimal duct (tear duct) which drains into the back of your nose and throat. Sounds like your eye drops could indeed be draining a bit into your nose and throat. You may get a taste of eye drops in your mouth, or a feeling that the drops are running down your throat. This is normal as the tear duct which drains tears to your nose will also drain some of the eye drop.. Doesn't sound like anything you have to worry about; it's quite normal and expected.
Then suddenly it goes away. Usually doesn't last more than a few minutes but very aggravating you can sometimes feel it coming. I only suffer from allergies and at least in my case it happens on the RIGHT side of my throat, and causes my RIGHT eye to water up when I cough. But left side is fine.