He knew that he was in danger of losing his head, may already have lost it, but submitted to the experience, even welcomed it.
The arrival of Ted Bishop, accompanied by his infant grandson, roused him from what may have been a brief trance.
There may even have been jealousy behind the iron closeness that united Fanny and her mother; neither was allowed to break their primitive agreement.
Anita Brookner, The Next Big Thing, chapters 10, 11, 17
Now reread those sentences. Is there a problem? I'm not so sure. Plainly they're in the past tense. And 'may' is certainly the present tense modal of which 'might' is the past tense version. Yes, yes. But should Brookner really therefore have written 'might' instead of 'may'? Many writers would, without misgivings, have written those sentences.
The problem, I think, is with the additional meanings or functions of 'might', i.e. its use not just as the past tense of 'may' but also as a means of expressing simple future intentions or possibilities in a language devoid of a future tense (I might go to the shops later), plus its common deployment in sentences expressing the hypothetical future or past (I might go to the shops later, if it doesn't rain / I might have gone to the shops if circumstances had been different). Use of 'may have' rather than 'might have' in sentences such as Brookner's above avoids these associations.
But here's Adam Mars-Jones on the subject, in a review that's not untypical of the way Brookner was received in some quarters during parts of her career:
The only sign of an awareness of contemporary language in The Next Big Thing is an unconscious one: for all her fastidiousness she succumbs to the confusion about 'may' and 'might'. He knew that he may have lost his head. He saw that she may have known. If her prose is to be lifeless, let it at least be correct.