.. Intuitiv, nach dem Klang ist "may not have been following them" einfach besser.

.. Das Einmalige "followed" trifft es ja nicht.

... Nochmals intuitiv: In welcher Weise sollte die Verneinung das Problem verändert haben?

people setting the rules may not have been following them

This communicates a simultaneity in the actions of setting and not following.

people setting the rules may not have followed them

This communicates consecutive actions, setting, then not following.

I am not sure which is worse, probably the simultaneity.

... Nochmals intuitiv: In welcher Weise sollte die Verneinung das Problem verändert haben?

Est ist wohl eher die Negation in verbindung mit dem "may", die mich verwirrt hat. Wenn es denn überhaupt stimmt, dass

"We have been following the rules"

ein Befolgen der Regeln über einen gewissen Zeitraum anzeigt, dann stellt sich mir die Frage, ob

"We may not have been following the rules"

in jedem Fall sagt, dass die Regeln möglicherweise über einen längeren Zeitraum nicht befolgt wurden, oder ob es auch bedeuten kann, dass bei Betrachten des längeren Zeitraums eventuell auch einzelne Verstöße zu beobachten waren. Letztere Möglichkeit ist vermutlich Unsinn, aber ich bin mir nicht ganz sicher.

harambee. Klar waren's nur einzelne Verstöße, soviele Gelegenheiten zu Corona-Partys gab's ja nicht, dennoch ist "continous" passend.

(Eigentlich schimpfe ich ja ungern selbst auf Österreich: Aber Gesetze gegen Korruption/fürTransparenz zu machen und selbst dagegen zu verstoßen passt zu oben)

The positive version would be "We may have been following the rules", which you can also word as "Perhaps we were following the rules"; this form is used to talk about something that was in the process of happening at a certain time in the past.

"We can use the present perfect progressive to talk about repeated actions and events, but not if we say how often they have happened (because this stresses the idea of completion - see above). Compare:

I've been playing a lot of tennis recently.

I've played tennis three times this week. "

Michael Swan , Practical English Usage, sec. ed. 1995, Oxford University Press

Ich als Nichtmuttersprachler hätte den Guardian Text auch so interpretiert, dass das Progressive wiederholte Verstöße betont. Aber das müssen Muttersprachler beurteilen.

It can be used to talk about repeated actions, yes. But the present perfect progressive does not have to describe repeated actions:

"I've been painting this wall since half past three" doesn't imply that you have had a break in that time.

What you have here, though, actually means "It is possible that the politicians were not following their own rules at that time".

Compare:

Definite: I live on the High Street.

Possible: I may live on the High Street. (Perhaps I live there. I don't want to tell you.)

Definite: I ate the apple yesterday.

Possible: I may have eaten the apple yesterday. (Perhaps I ate the apple yesterday. I don't want to tell you.

Definite: In May 2020 I was following the Covid rules. (= This was the principle I usually followed at that time.)

Possible: In May 2020 I may have been following the Covid rules. (= Perhaps this was the principle I usually followed at that time. I don't want to tell you.)

We may not have been following the rules.

Eine andere sagt:

True, you have not been following the rules.

That conversation makes no sense. The first person is talking about an activity in progress in the past, at a time that is now historical. The second person is talking about a time continuing up to the present or that is very recent.

Nebenbei:

Wenn's recht ist, werde ich mal eine Tippfehlerkorrektur für die Betreffzeile machen (im Text steht es auch nochmal falsch).

Nicht: continous, sondern continuous 

Denk einfach an: kontinuierlich. Du schreibst und sagst ja wohl auch nicht: kontinierlich :)

Oder du schreibst einfach progressive :-)

Haben die Beispiele in #8&9 es eigentlich klarer gemacht?

Denk einfach an: kontinuierlich. Du schreibst und sagst ja wohl auch nicht: kontinierlich :)

Das Problem, das wohl auch andere kennen: Wenn ich darüber nachdenke, dann weiß ich ja, wie es richtig ist. Aber meine Nachdenkressourcen sind endlich, deswegen muss ich sie gut einsetzen.

Haben die Beispiele in #8&9 es eigentlich klarer gemacht?

Ehrlich gesagt nein. Sie haben mich erstmal noch mehr verwirrt, aber ich weiß auch nicht, was ich fragen könnte. Gerade #9 hat irgendwas mit meinem fehlenden Verständnis zu tun, aber ich kann es nicht auf den Punkt bringen.

Sorry, I don't know what the right name is for this construction with "may" or I might find something that could explain it better. I'll try to explain it again, but I'm not sure if I can do it any more clearly. Maybe someone else knows what it's called.

In June 2020, Boris Johnson held parties at Downing Street.

At that time, most people were following the Covid rules such as not meeting more than 2 people indoors. Sie haben sich zu dem Zeitpunkt an die Regeln gehalten.

People are angry that perhaps the politicians were not following the rules in June 2020.

= People are angry that politicians may not have been following the rules in June 2020.

(Those last two sentences mean the same thing, dass sie sich eventuell zu dem Zeitpunkt nicht an die Regeln gehalten haben.)

The present perfect progressive e.g. "they have been following" is normally related to the present in some way. It means that the time you are describing is not finished now, or has only just finished now, in the present.

The past progressive e.g. "they were following" is not related to the present. It's talking about an activity that was in progress at a finished, historical time in the past.

When you add "may" to something in the past progressive, meaning "perhaps", it changes the form of the sentence so it looks like the present perfect progressive. But it's still talking about the past only; it isn't related to the present at all, and doesn't work like the present perfect progressive normally does.

Danke sehr, CM2DD, das macht es zumindest etwas klarer!

Zum Satz

1. People are angry that politicians may not have been following the rules in June 2020.

Der enthält ja eine definierte Zeitangabe, bei der man in den allermeisten Fällen das present perfect besser vermeidet. Muss man das present perfect progressive da nicht vermeiden oder liegt es hier am "may not"? Anders gefragt: Muss ich

2. Politicians have not been following the rules in June 2020.

weniger vermeiden als

3. Polticians have not followed the rules in June 2020.

oder ist der Satz 2 ebenso falsch wie Satz 3 und ist nur Satz 1 korrekt?

There is no present perfect continuous in the original sentence. Tenses of non-finite forms like the infinitive (without to) have been following after may are not absolute but relative to the finite form of the verb in the sentences, i.e. recognises or are. If the action expressed by the infinitive and the finite verb form in the sentence are simultaneous, the (present or simple) infinitive is used (e.g. to follow or to be following). If the action expressed by the infinitive takes place prior to the action expressed by the finite verb form in the sentence, the perfect infinitive is used (e.g. to have followed or to have been following).