Is fasting on Tuesday, which is the day of Eid in the questioner's country, forbidden if the Day of 'Arafah in another country was on Monday?

The scholars' disagreement is well-known regarding the issue of the رؤية (sighting) by people of a single locality, and whether it applies to all lands or not. This disagreement extends to all months, including Dhul-Hijjah. What is authoritative for us is to consider the اختلاف المطالع (difference in moonrise/sighting horizons). Therefore, if your country relies on the شرعية رؤية (legitimate sighting) for establishing the beginning of the month, then act according to the رؤية of your own people, even if it differs from the رؤية of the people of Mecca. There is no harm in this, according to the opinion that considers the اختلاف المطالع.