Based on the material I've read up to this point, Italy was a significant liability to Germany, at least militarily. In Greece and North Africa, the Italians got pummeled and required Germany to divert military forces to bail them out.

All true. It forced several interventions from Germany, in Greece, in North Africa after Operation Compass, etc. Italy's conduct disastrously gave the British the opportunity they had been bereft of since Dunkirk; the ability to fight and destroy Axis units on the ground.

So what, if any, benefit was there for Germany to join forces with Italy?

The Italian ground forces are disparaged constantly, and not without reason, in WWII. Like all militaries, the Italians had their share of premier units and instances of professionalism and bravery under fire. However the fact remains undeniable that the Italians were simply not prepared for World War II.

That does not mean however that the Italian ally was a corpse that the Germans were shackled to. Asides from committing an Army to their Army Group South in Operation Barbarossa, the Italian Navy provided essentially the only surface fleet of note in the Mediterranean. Once the German efforts in North Africa ramped up considerably, its no exaggeration that the efforts of the Italian navy kept the fight in Africa alive. Now, I'll temper this praise with the fact that if it wasn't for Italian unprepardness, the Germans never would've had to enter the North African theater to begin with, but I digress.

Let's look at this more in depth.

So, perhaps most importantly, lets think what the man on the spot thought of his Italian allies: Rommel. Rommel chafed under Italian command, and his thirst for mobile operations often brought him to a head with his Italian 'superior' - Bastico. The relationship was so acidic that Rommel frequently referred to his counterpart as "Bombastico." Rommel's contempt for the Italian command expanded past his immediate chain of command, however:

...among Rommel's remarks there are also claims against the High Commands in Rome....and open charges of treason against the Italian Navy Officers who 'sabotaged whatever they could...'

(Page 179)

The author though, wisely chooses to look at actual factoids. Its very clear that the efforts of the Italian navy to keep Rommel supplied can be described as anything but 'treasonous.' Given the sparse number of German surface boats, the bastion of Gibraltar, and the strain the Ju-52 air-transport fleet was putting itself under, between operations in Yugoslavia, Russia, Crete, etc. we can conclude that the prime supplier of the Deutches Afrika Korps was, in fact, the Italian navy.

The numbers being supplied ebbed and flowed to an extent with the efforts of the Royal Navy and the effectiveness of sorties from Malta, but remained stable and relatively plentiful between 1941-42. The numbers between April-May 1942 are accessible to us:

...the Axis were able to bring to Africa in the month of April some 150, 389 tons of supply and equipment of the 151, 578 tons shipped..."

Whether this tonnage was sufficient to sustain offensive operations is highly debatable, but the fact remains: of the tonnage that was shipped, the vast majority made it to shore intact. This is evident of a great show of effort on part of the Italian Navy to supply not only its own 10th Army, but the German Afrika units. The source goes on, detailing the following months:

...against 47, 588 tons arriving in March. May was also a good month: 86, 439 tons arrived of the 93, 188 shipped. The combined Axis effort at sea and in the air had helped...

(Page 145)

My emphasis. You can see losses were being suffered given the tonnage loss in May. So its obvious that not only was the Italian Navy making a repeated effort to keep itself and its ally supplied, but it was doing so at cost to itself across a hostile and contested stretch of sea. Rommel's opinions on this matter should therefore be discounted near entirely.

The amazing part of all this is, the Italian navy was not prepared to conduct such operations in the Mediterranean and adapted remarkably well:

Ironically, the Italian navy was informed before the war that resupplying the African colonies was not their strategic duty...but with the fall of France, suddenly the navy had a new mission to supply Africa... a mission which it performed well over the next three years, failing for a few short key periods..."

(Page 20)

Source: Rommel's North Africa Campaign: September 1940 - November 1942 by Greene, Jack et. Massignani, Alessandro.