may have + PAST PARTICIPLE

In the English Grammar Profile, B2 point 163 MODALITY is defined as:

‘may’ in phrases such as ‘as you may know’, or ‘as you may have’ + ‘-ed’ to focus the reader on shared knowledge.

This partly overlaps point 168 MODALITY and  PAST AFFIRMATIVE of

‘may have’ + ‘-ed’ to talk about possibility in the past.

Also note the entry in the English Vocabulary Profile:

may have = B2 used to talk about things that possibly happened in the past or might possibly have happened by a certain time in the future

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A search for collocates of ‘may have’ in COCA:

Past possibility example:

He said he believes drug use or mental illness may have contributed to Carter‘s behavior, but that hasn’t been confirmed.

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A search on iWeb for as you may * * * *:

1 AS YOU MAY OR MAY NOT KNOW 950

As you may or may not know,

 big things are happening here in Williamsburg, Brooklyn.

2 AS YOU MAY HAVE NOTICED , THE 288

As you may have noticed,

 the chat function has disappeared 

from the top of the Forum.

3 AS YOU MAY HAVE GUESSED , THE 227

4 AS YOU MAY HAVE NOTICED , I 201
5 AS YOU MAY HAVE NOTICED , WE 194
6 AS YOU MAY BE ABLE TO TELL 189
7 AS YOU MAY NOT BE ABLE TO 189
8 AS YOU MAY HAVE GUESSED FROM THE 184
9 AS YOU MAY ALREADY KNOW , THE 181
10 AS YOU MAY HAVE ALREADY GUESSED , 171
11 AS YOU MAY BE AWARE , THE 159
12 AS YOU MAY KNOW BY NOW , 143
13 AS YOU MAY HAVE HEARD , THE 136
14 AS YOU MAY OR MAY NOT BE 130
15 AS YOU MAY HAVE GUESSED , I 127
16 AS YOU MAY HAVE GUESSED , THIS 125
17 AS YOU MAY OR MAY NOT HAVE 113
18 AS YOU MAY HAVE GUESSED BY NOW 112
19 AS YOU MAY HAVE ALREADY NOTICED , 106
20 AS YOU MAY ALREADY BE AWARE , 102

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*Notice the adverb ‘already’ is quite common in the middle position.  For example:

I need to know if they may have already happened.

listen

Here are the results of an iWeb search for the most common adverbs in the middle position with the most common past participles:

may_VM have _RR _VVN

1 MAY HAVE ALREADY SEEN 781
2 MAY HAVE NEVER HEARD 628
3 MAY HAVE ALREADY NOTICED 608
4 MAY HAVE ALREADY HEARD 501
5 MAY HAVE ALSO NOTICED 385
6 MAY HAVE NEVER SEEN 322
7 MAY HAVE JUST FOUND 317
8 MAY HAVE NEVER THOUGHT 289
9 MAY HAVE ALREADY GUESSED 262
10 MAY HAVE JUST GOTTEN 224
11 MAY HAVE ALREADY STARTED 219
12 MAY HAVE FINALLY FOUND 203
13 MAY HAVE EVEN SEEN 193
14 MAY HAVE ALSO SEEN 190
15 MAY HAVE ALSO HEARD 159
16 MAY HAVE NEVER CONSIDERED 151
17 MAY HAVE ALREADY BEGUN 136
18 MAY HAVE ALSO CONTRIBUTED 130
19 MAY HAVE ALREADY TRIED 128
20 MAY HAVE NEVER KNOWN 127