Is Breaking Fast Before Adhan Allowed?
Question

Is it permissible to eat a few seconds before hearing the adhan, knowing that I cannot hear the adhan and the neighbourhood is Shi’ah and they give the adhan after we do?
Praise be to Allah.When to break your fast
When the sun has set, it becomes permissible for the fasting person to break the fast , whether the muadhdhin has given the adhan or not. What counts is the setting of the sun. Prophet Muhammad (peace and blessings of Allah be upon him) said: “When the night comes from here and the day departs from here, and the sun has set, then it is time for the fasting person to break his fast.” (Narrated by al-Bukhari, 1954; Muslim, 1100) 
Ibn Daqiq al-‘Eid said: “This hadith is a refutation of the Shi’ah who delay breaking the fast until the stars appear.” (Fath al-Bari)
Some muadhdhins may delay the adhan until a while after the sun has set, so the adhan is not what counts. This action is contrary to the guidance of the Prophet (peace and blessings of Allah be upon him) who urged us to hasten to break the fast after the sun has set. He said: “The people will still be fine so long as they hasten to break the fast.” (Narrated by al-Bukhari, 1957; Muslim, 1098)
It is permissible for the fasting person to break the fast if he thinks it most likely that the sun has set; it is not essential that he be certain, rather thinking it most likely is sufficient. 
If the fasting person thinks it most likely that the sun has set, and he breaks his fast, there is no sin on him. But it is not permissible for him to break his fast if he is not sure whether the sun has set. 
Can you break the fast when you think it most likely that the sun has set?  
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said: 
It is Sunnah to hasten to break the fast when the sun has set. What counts is the setting of the sun, not the adhan, especially in present times when the people rely on timetables and they follow the timetables based on the clocks, and their clocks may be fast or slow. If the sun sets and you see it, and the people have not yet given the adhan, then you may break the fast. If they give the adhan and you see that the sun has not set, then you should not break the fast, because the Messenger (peace and blessings of Allah be upon him) said: “When the night comes from here and the day departs from here, and the sun has set, then it is time for the fasting person to break his fast.” 
It does not matter if there is still strong light. Some people say: We will carry on fasting until the disk of the sun has disappeared and it has gotten somewhat dark. But this does not matter; rather you should look at the disk of the sun and when the top of it disappears, then the sun has set, and it is Sunnah to break the fast. 
The evidence that it is Sunnah to break the fast at this time is the words of the Prophet (peace and blessings of Allah be upon him): “The people will still be fine so long as they hasten to break the fast. ” Hence we know that those who delay breaking the fast until the stars appear, like the Rafidis, are not fine. 
If someone were to say: Can I break the fast when I think it most likely that the sun has set?  
The answer is yes, and the evidence for that is the report that is narrated in Sahih al-Bukhari from Asma bint Abi Bakr (may Allah be pleased with her) who said: We broke our fast on a cloudy day at the time of the Prophet (peace and blessings of Allah be upon him), then the sun appeared. It is known that they did not break their fast on the basis of certain knowledge, because if they had broken their fast on the basis of certain knowledge the sun would not have appeared after that; rather they broke their fast on the basis of their thinking it most likely that it had set, then the clouds dispersed and the sun appeared.” (Al-Sharh al-Mumti’, 6/267)
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And Allah knows best.