Es gab kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar
Es soll kurz vor der Trauung eine Beziehungkrise bei dem Prinzenpaar gegeben haben
What is the tense of the transformed sentence? Secondly why is it no longer prateritum?
Es gab kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar
Es soll kurz vor der Trauung eine Beziehungkrise bei dem Prinzenpaar gegeben haben
What is the tense of the transformed sentence? Secondly why is it no longer prateritum?
Es gab kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar
Es soll kurz vor der Trauung eine Beziehungkrise bei dem Prinzenpaar gegeben haben
First off: these two sentences mean different things. The first one states some fact (the two had a crisis shortly before their marriage). The second one states that there is a rumor about some crisis they had.
"Es soll ... gegeben haben." is a phrase which means "Allegedly there was [something]."
So, the first sentence talks about a fact (the crisis) and this fact happened in the past. This is why the sentence is in Präteritum ("es gab"), which is the tense for relating things from the past ("Erzählform").
The second sentence is about two things: one is said crisis, but the main fact is the rumor about it. The content of the rumor may be in the past (hence: Perfekt, "gegeben haben"), but the rumor is now (hence: Präsens "soll").
The purely present version would be
Es soll eine Krise geben.
"There supposedly/reportedly is a crisis."
Note the construction "geben sollen" and that in the complete sentence "geben" is an infinitive. We then cannot simply put "geben" in past tense as the point of an infinitive is not to have a tense. However, Perfekt, which also describes the past, is formed with an auxiliary, "haben", so we can use that:
Es soll eine Krise gegeben haben.
Note that "haben" is again an infinitive.
A purely formal perspective can sometimes be useful to understand such examples.
Es [geben + PAST REFERENCE] damals eine Krise.
Es soll damals eine Krise [geben + PAST REFERENCE].
Both sentences require a form of geben that makes reference to the past. In the first example, the simple past gab is a valid choice to make that reference; however, the modal soll in the second example requires an infinitive and therefore the infinitive perfect gegeben haben is used. The time reference of geben doesn't change between the examples; only the tense used to express it.
Whoever created that exercise sheet is a Northern speaker. Northern speakers treat the auxiliaries, the modals, and some other very common verbs slightly different. Geben is one of those verbs.
They commonly use Präteritum as a shortcut for Perfekt for those verbs.
Es gab kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar.
Es hat kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar gegeben.
Those are equivalent in northern German. And that explains why the phrasing with sollen becomes:
Es soll kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar gegeben haben.
This is the modal Perfekt tense in which haben/sein is put as the infinitive complement of the modal, and the Partizip II of the original Perfekt expression is retained.
This combination Partizip II+haben/sein is also called Infinitiv II or Infinitiv Perfekt. And there is also a Partizip II+zu+haben/sein form of that.
Note there is a second Perfekt tense with modals:
Es hat kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar geben sollen.
This is the "normal" Perfekt which treats the modal as any other verb. It means a different thing. This "normal" Perfekt moves the viewpoint of the modal expression to the past. So the modal evaluation happened back then. In this particular context, it reads as if you wished they had a relationship crises back then. Compare the non-Perfekt sentence:
Es soll kurz vor der Trauung eine Beziehungskrise bei dem Prinzenpaar geben.
That's a wish.
If that gab was a true Präteritum meant for storytelling, the soll phrasing indeed had to be:
Es sollte vor der Trauung eine Beziehungskrise bei dem Prinzenpaar geben.
But the author of the sheet did not tell a story. She's simply a Northerner.
The verbs for which this frequently happens are haben, sein, werden, bleiben, müssen, dürfen, können, sollen, wollen, mögen, bekommen, gelten, heißen, wirken, geben, kommen, gehen, fahren. Plus their prefixed variants. And this list is not exhaustive. The more northern the speaker is, the more verbs go on their personal list.
No, this is not some "Northern" thing. It's a subjective usage of a modal verb to show that the person was not there, but heard about it. In this case we use "wollen" (when the person said something themselves) and sollen (when we hear about something or someone). For example:
I say "Ich hasse Orangen."
A reporter would then say "Er will Orangen hassen." This does not mean "he wants to..." but rather "he said this about himself and I am reporting it."
If you heard/read "Am Samstag gab es eine Veranstaltung..." then a repoter would say "Es soll am Samstag eine Veranstaltung gegeben haben."
These usages of indirect speech called subjektiver Gebrauch der Modalverben are always in Präsens (the conjugated verb is at least). We differentiate the time by showing that with the verb, like "es soll...gegeben haben" if the event already took place or is going to (es soll...geben). The same with wollen "Sie will...sagen/gesagt haben."
There are also helpful YouTube videos and websites written about this topic, like: https://deutschegrammatik20.de/spezielle-verben/modalverben/subjektive-modalverben-einfuehrung/
Or: