After Rommel's second defeat of the British in North Africa around Tobruk that city became sieged once again. The first time this happened the Australians defending Tobruk successfully held off the Germans. This time, the city fell after a several week siege.
What if Rommel instead of investing Tobruk had pushed the Afrika Korps east after the retreating British and left primarily the Italians to just invest and pin the defenders in Tobruk?
Instead of the British getting several weeks of respite to build the el Alamein position they are hounded on their heels by the Germans. I suspect that the British would have proved unable in such conditions to establish a defense at Alamein and in not being able to do so would have lost Egypt.

Off the top of my head, I would think that an unconquered Tobruk, given it's location, would quickly become a major thorn in the supply chain of Rommel's Afrika Korps. I would imagine that there was at least one useable airstrip there to launch British air strikes from.

I´d recall after the Gazala battle the Germans ( i.e. Rommel ) pushed his tired troops towards Tobruk because he knew he had to be there fast. That´s why I have thought the city fell soon.

"Just seven days later, on 21 June 1942, in circumstances that even with the benefit of a subsequent formal court of enquiry remain obscure and contradictory, 35,000 Allied troops (including the entire South African 2nd Division), were surrendered to General Enea Navarrini's 30,000 troops."

Battle of Gazala - Wikipedia, the free encyclopedia

In that sense I´d think just having enough powerful troops would, Italian troops would do, and Tobruk would probably fall. But whether the German troops would be in condition good enough to continue eastwards straight away, I´m not sure.

One must also remember that about Auchinleck : Here Auchinleck tailored a defence that took advantage of the terrain and the fresh troops at his disposal, stopped the exhausted German/Italian advance in the First Battle of El Alamein ( 1–27 July 1942 ).

First Battle of El Alamein - Wikipedia, the free encyclopedia

So Rommel did attack one week after Tobruk fell eastwards.

Later on Auchinleck was kicked out because he was not believed to be able to win the war. However I think some sources claim that Monty´s victories were prepared by Auchinleck.

I´m sure we´ll discuss this more later on. Any new info/ideas are welcome! This is just how i recall the situation.

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Churchill was disapponted when Auchinleck reported he could not make a large scale and decisive offensive before late August or more likely September. Auchinleck specified the additional reinforcement he needed before he could attack and pointed out those could not be ready before the end of August.

General Gott was choosen to replace Auchinleck. It is claimed he was no more optimistic about a july or August attack than Achinleck. Gott had been a reasonablly sucessfull corps commander and certainly had experince in the mobile warfare of Africa. Gott was killed in a aircraft crash before he could do anything at all.

Montgomery appeared to agree to everything. Then after taking command he infuritated Churchill by demanding even more reinforcement than Achinleck and stating he could absolutely not attack untill October. Some say Monty wanted to wait until November, when the Torch operation was executed.

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It is that week of delay that is critical. Had Rommel immediately pursued the British into Egypt and just invested Tobruk he likely would have pushed them back to the outskirts of Cairo and Alexandria. Had this happened, Tobruk would fall on its own.
Alexandria would be abandoned as a naval base taking all the shipping and equipment that wasn't nailed down (and alot of what was nailed down) with it. This would have left Tobruk with just what supplies were on hand and with none coming.
Even if the British eventually are able to fend off the Germans their not having a sound position on which to anchor a solid line would have been sufficent to allow the Germans much of the advantage they needed.
The big question in this is could Rommel hold the ground? With his supply lines even more tenious than they were historically could he take Alexandria? If so his supply problems are largely over and he wins for the short term.
The US could still land behind him as they historically did. This would end any further moves East as Rommel would have to deal with the US troops to his rear. But, in the interm he could have wreaked havoc on the British in Egypt.

Definitely I agree that Rommel´s time window was after Gazala to win all he can in the fastest possible time. I recall Rommel forced his troops to attack towards Tobruk because he knew it would save alot of German blood.

However, is it possible Rommel also needed this to move ahead again?

"Tobruk fell on June 21, and the Axis forces captured 2.5 million gallons of much-needed fuel, as well as 2,000 wheeled vehicles."

World War II: North Africa Campaign » HistoryNet - From the World's Largest History Magazine Publisher

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I like what von Rundstedt posted (the serious one, of course.) However, TA did point out that Rommel had a supply problem. Would consolidating his available supplies for a smaller mobile force to pursue the British be enough to overtake the British rear? It's possible but I have to doubt that this would be strong enough. It could harass the British but then this mobile force when in contact would be open to air attack.
von Rundstedt did pose a good possibility though. I have to admit that even if Spain just went through a civil war, it's within the realm of possibility that a German success in taking Egypt could entice the Spaniards to join the Axis.

In the end, the strain on Rommel's supply line was his achilles heel. Besides, that pesky little corporal in Berlin annoyed Rommel so much that I believe it clouded Rommel's judgement.

von Rundstedt, nice explanation, as usual. Rommel did have the potential to run roughshod over the Allies but The Allies had an ace that Rommel didn't have, which was ULTRA.
I have to concede, though, that ULTRA does have a limitation. With Rommel in this scenario highly mobile, the Allies might not be able to disseminate the needed intelligence information fast enough to the commanders in the field. Therefore, it's possible that even with the Allies having ULTRA, Rommel might succeed with a healthy dose of supplies and a lot of luck.