This is the 5882nd most frequent Russian word.
реальную
"Real" or "actual" (feminine singular form in Russian).
The word 'реальную' here is the accusative singular form of the adjective 'реальный' (real) and agrees with the feminine noun 'историю' (story), describing its quality as being 'authentic' or 'true'.
Она сказала, что пишет статью, основанную на реальную историю.
She said she is writing an article based on a real story.
'Реальная' is the nominative singular feminine form of the adjective 'реальный' (real), which attributes true existence or significance to the feminine noun 'возможность' (opportunity).
У нас была реальная возможность выиграть в соревновании.
We had a real chance to win the competition.
Here, 'реальную' in the accusative singular form agrees with the feminine noun 'картину' (depiction), denoting the characteristic of being based on reality or being accurate.