real
Actually existing, corresponding to reality and not merely imagined, invented, or theoretical.
Die Gefahr eines Stromausfalls ist durchaus real.
The danger of a power outage is quite real.

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Actually existing, corresponding to reality and not merely imagined, invented, or theoretical.
Die Gefahr eines Stromausfalls ist durchaus real.
The danger of a power outage is quite real.

In its basic meaning, 'real' is often used with abstract nouns to emphasize that they actually exist. Typical combinations are 'eine reale Gefahr' (a real danger), 'eine reale Chance' (a real chance), or 'ein reales Problem' (a real problem).
Es besteht eine reale Chance, das Projekt noch zu retten.
There is a real chance of still saving the project.
You can recognize the meaning 'adjusted for prices' from the context. It is used almost exclusively in economic contexts with words like 'Einkommen' (income), 'Lohn' (wage), 'Zins' (interest), or 'Wachstum' (growth).
Die Zinsen sind zwar hoch, aber der reale Gewinn ist nach Inflation gering.
Interest rates are high, but the real profit after inflation is low.
The meaning 'authentic' or 'honest' is typical of colloquial and youth language. It is rarely used in formal or written contexts.
Er versucht immer, cool zu sein, aber das wirkt einfach nicht real.
He always tries to be cool, but it just doesn't seem real.
really
'wirklich' and 'real' can both express that something is not just imagined, invented, or apparent. 'wirklich' is very flexible in German and can be used as an adverb or an adjective. 'real' is mainly an adjective and fits well when you want to emphasize that something actually exists or occurs in reality in a practical sense. 'wirklich' often sounds more natural in everyday language, especially for emphasis. 'real' feels more precise when the focus is on the difference between imagination and reality.

Nach dem Gespräch wirkt das Problem viel realer als vorher.
After the conversation, the problem seems much more real than before.
Er hat die Aufgabe wirklich verstanden, weil er sie gut erklären konnte.
He really understood the task because he was able to explain it well.