The Germans and Japanese were part of the Axis as of September 1940. As part of the the Tripartite Pact, Berlin ceded "Greater East Asia" to Japan.
Shortly before the signing of the Pact, the Japanese had invaded the Vichy French colonies in Indochina. Since Vichy France was under the control of Berlin, the Japanese invasion seems like it should have been tantamount to an attack on German territory.
What is the backstory to this? Why did the Germans recognize Japanese "leadership" in the region, and what did the Germans get in return? Was there public outcry in Germany at the loss of Indochina, or were those colonies seen as French instead of German?
Thanks!
From Wikipedia:
In 1940, during World War II, the French were defeated by the Germans. The French State (commonly known as Vichy France) was established as a Client state of Nazi Germany. The French colonial authorities, in French Indochina, sided with the Vichy regime. In September 1940, Japan invaded Indochina. Following the cessation of fighting and the beginning of the Japanese occupation, the French colonial authorities collaborated with the Japanese. The French continued to run affairs in Indochina, but ultimate power resided in the hands of the Japanese.
(Wikipedia cites Jonathan Neale's The American War: Vietnam, 1960–1975 for this paragraph.)
There are a few misconceptions in your post that I feel should be addressed.
First off, Germany and Japan were natural allies, given that they were both opposed to by the established powers in their respective regions (France and Britain) and they both had an intense hatred of the Communist USSR. Germany didn't have the ability to project force in East Asia at all, so they didn't really "cede" that land to Japan, they more just acknowledged that Japan would be the dominant power in Asia, while Germany ruled in Europe.
Secondly, Germany didn't "own" Vichy France, Vichy France was an independent state and the direct continuation of the old French government. Now Vichy France collaborated with Germany, but at no point were any of Vichy's colonies given to Germany, nor was Vichy territory seen as "German".
So therefore the Japanese invasion of French Indochina was perfectly within there rights according to the Tripartite pact, and Germany didn't lodge any complaints.
Essentially the Germans had a very exploitative and self-serving attitude towards the European colonies of occupied Europe. If Germany could not wring any immediate material gain from them, Hitler was indifferent to their fates.
Vichy was in something of a unique state compared to occupied Europe in that its government did not go into exile. Therefore, in theory, its colonies were still subject to the metropole's political control. Various entities within the Vichy French colonial service attempted to articulate a vision in which French colonial space would coexist and supplement the new order promised by the Third Reich. One of the most prominent of these advocates was Edmond Vivier de Streel, an industrialist and former cabinet chief of the Colonial Ministry. Vivier de Streel envisioned France as the colonial power within a German-dominated Europe; it would be the standard-bearer of European civilization and act as a source for colonial products for Europe. In the shadow of defeat, Vichy emphasized its colonial presence as evidence of French cultural strength and a sign for its spiritual regeneration under Vichy's auspices.
The Japanese invasions were a rude shock to the Vichy colonial pretensions. Paul Baudouin, the Vichy foreign minister, was apoplectic about an Asian power occupying a European colonial power. His appeals to Berlin to rein in Japan fell on deaf ears as strengthening Japan was geostrategically beneficial to Germany. Like in 1940, the overwhelming military force led to an uneasy collaboration; the Japanese saw that keeping the existing French administration intact (albeit under close supervision) was a beneficial expedient. Ironically, Indochina remained the last functioning European colony east of India during the war.
Sources
Jennings, Eric T. Vichy in the Tropics: Petain's National Revolution in Madagascar, Guadeloupe, and Indochina, 1940-44. Stanford, Calif: Stanford University Press, 2001.
Lackerstein, Debbie. National Regeneration in Vichy France Ideas and Policies, 1930-1944. Farnham, Surrey, England: Ashgate, 2011.