What is the ruling on fasting a day with the intention that it is the tenth of Muharram according to the available calendar, then discovering after midday that it is actually the eleventh of Muharram?

You should have investigated and inquired about the time of sighting the crescent moon according to Islamic law (Shari'ah). What is considered valid is the Shari'ah sighting, not the calendar. If you did not fast the tenth because you believed it was the ninth, then the Sunnah of fasting Ashura has passed, but you are rewarded for your sincere intention, as the Prophet, peace and blessings be upon him, said: "If a servant falls ill or travels, Allah records for him what he used to do when he was healthy and resident." You will be rewarded for fasting the eleventh, and you should investigate carefully in the future.