The question is on the highlighted sentence in this excerpt from L'Étranger by Camus.
J’ai dit : « Il est tard. » Raymond le pensait aussi. Il a remarqué que le temps passait vite et, dans un sens, c’était vrai. J’avais sommeil, mais j’avais de la peine à me lever. J’ai dû avoir l’air fatigué parce que Raymond m’a dit qu’il ne fallait pas se laisser aller.
I understand that it means, "I must have looked tired."
What would this mean: Je dois avoir eu l’air fatigué.
How does one say in French: "I had to look tired." (Imagine that you, wanting to stay in for the evening, had told your friends you were too tired to go out. But that only brought them to your house, and now you had to look the part.)
You don't have to read this background to answer the question above. Possibly it may only cause confusion.
Consider these two groups of sentences.
(D1) Ich musste müde aussehen.
(E1) I had to look tired.
(F1) J’ai dû avoir l’air fatigué.
(D2) Ich muss müde ausgesehen haben.
(E2) I must have looked tired.
(F2) Je dois avoir eu l’air fatigué.
If we only had the German and the English sentences, we may say that Group 1 was all about the past. Both the looking (seeming, appearing) and the modality are in the past. But in Group 2, only the looking is in the past while the modality is in the present. Group 2 is about the speaker's present estimation of his past appearance.
But apparently, French does not work like that. F1 is in meaning a present modality about a past condition even if its surface grammar suggests something else, namely a past modality about a coeval (i.e. past) condition (state, event).
We may say that F1 has taken on F2's job. Which leads to the questions 1 and 2 as above.
Also I am worried about sentence pairs like these (which may not even be well formed sentences):
(F1') J'ai pu le dire.
(F2') Je peux l'avoir dit.
Before encountering F1 in Camus, I may have thought (guessed) F1' meant, "I was able to say that (e.g. fortunately)" and F2' "I could have said it (e.g. but who knows, I was drunk)." But now I wonder whether F1' also steals the job of F2'.