I think that the Inspector gives it away when he gets far too emotional and worked up about things. A real police inspector would not get so involved. Here is an extract from the play, at the end of Act 2 where the Inspector gets too involved:
Inspector: "(very sternly) Her position now is that she lies with a burnt-out inside on a slab. (As Birling tries to protest, turns on him.) Don't stammer and yammer at me again, man. I'm losing all patience with you people. What did he Say?
Here the Inspector gets too emotional about such a small thing. I also think that a real police inspector would treat Birling with more respect. It is true what Birling says about him being a public man and telling the chief of police because he is a friend. Here is an extract where the Inspector treats Birling with disrespect; it is near the beginning of Act 2:
Sheila: (urgently, cutting in) You mustn't try to build up a kind of wall between us and that girl. If you do then the Inspector will just break it down. And it'll be worse when he does.
Mrs B: I don't understand you. (To Inspector.) Do you?
Inspector: Yes. And she's right.
Mrs B: (haughtily) I beg your pardon!
Inspector: (very plainly) I said Yes-I do understand her. And she's right.
Mrs B: That, I consider to be a trifle impertinent, Inspector.
Here the Inspector does not treat Mrs Birling with the respect a police inspector would.
I think there is a possibility that the Inspector could be Eva Smith and just goes back to haunt them. This maybe quite far fetched by I would not rule it out completely, because if it were true then it would explain everything.
Another quite reasonable explanation would be that the "Inspector" actually phoned them up and pretended to be the police just to give them one last scare.
I think that a real Inspector would leave far more formally rather than just storming out:
Inspector: But just remember this. One Eva Smith has gone, but there are millions and millions and millions of Eva Smiths and John Smiths still left with us, with their lives, their hopes and fears, their suffering and chance of happiness, all intertwined with our lives, and what we think and say and do. We are responsible for each other. And I tell you that the time will soon come when, if men will not learn that lesson, then they will be taught it in fire and blood and anguish. Good night.
He walks straight out, leaving them staring subdued and wondering. Sheila is still crying. Mrs Birling has collapsed into a chair. Eric is brooding desperately. Birling, the only active one hears the front door slam, moves hesitatingly towards the door, stops, looks gloomily at the other three, then pours himself out a drink, which he hastily swallows.
Here I firstly think that he leaves far too informally to be a convincing Inspector. Secondly I think that when he is coming to the end of his final speech he becomes too emotional to be a real person. I feel he almost knows what is to come and as if he does this a lot to people.
In conclusion I think that the Inspector is as real as every one else in the play but I think that he represents justice or truth and is a form of angel or something along those lines.
The Inspector's movements on stage are swift and straight, almost haunting (perhaps this is why his name is Goole? There is a possibility that the Inspector is indeed associated with the world of the strange and unpredictable. Why not give him a name associated with such ghostliness?). He would wear a long green trench-coat but nothing to signify that he was from any sort of police branch. His tone of voice should be deep, serious and grave. His relationships with the other characters should be as follows:
Mr Birling - cold and harsh. Mr Birling is a stereotypical middle/upper class man (Pompous, haughty, possibly rather fat, etc.). He would think that, being a man of his status, he would be all of the above. This would make him select a figure and posture that dominated Birling and make him fearful of him. His tone of voice would be barking, stern and harsh.
Mrs Birling - impatient and uncaring. He would think, similarly to as he did with Birling, that she would be the stereotypical middle/upper class woman and that she would behave alike to her husband. He would stand over her to dominate her feelings and not give her time to think. He would use a harsh, unsympathetic tone of voice.
Gerald - slightly more considerate. He would look upon him as a wealthy young middle/upper class man. He would realise that this young man has had no experience in this sort of life's troubles. He would have more lenience towards him than he would to Mr and Mrs Birling. He would use a rough voice to get his message across.
Eric -straightforward and lenient. He would look at him and see a frightened young man who is obviously traumatised by the whole experience and has never been through anything as tragic as this event in his life. He would use a careful, steady voice
Sheila - very caring and. He would think of her as a sad, inexperienced girl who deserves looking after. He would realise that because she has been a bit spoilt through her short life, she would have had no idea about just how important her comments to Eva Smith were at the time. He would use a gentle, soothing voice to get through to her and coax her thoughts and truths out of her.
In this play it appears that for each person, the Inspector adopts a different posture and tone for each character in the play, even if they come one after the other in a string of long speeches.
The audience reacts to the Inspector as though he was in fact some sort of premonition. His character would be very mysterious and leave his audience thinking about who or indeed what he really was until long after the performance. This is exactly the kind of atmosphere Priestley wished to create on the set. I think you will agree with me when I say that the Inspector is one of the most shrouded and mysterious figures ever seen on stage.
Please find enclosed, a marked copy of the playscript. I hope you can be present at the production and that you will enjoy your time with my fellow theatre workers and myself. I look forward to meeting you at the first rehearsal.
Yours sincerely,
Henry H. Dean (Director)