Waving his flag around the globe?

Question

I'm onto translating an article about a chef who has several restaurant around the globe and the original sentence (Turkish) included an idiom like following to indicate the chef's omnipresence and popularity in the culinary scene:

"Thus he's currently waving his flag around the globe."

or in a more direct translation:

"He has his banners waving in every corner of the world."

Yet, I cannot translate the idiom directly and I don't have any alternative to use instead.

Can you help me find the closest translation?

Answer 1

The chef in question is making his presence known by ... waving his flag or his banners [promoting himself or advertising his products] [widely] around the globe or in every corner of the world.

Answer 2

I think you could possibly maintain the flag/banner notion by changing "wave" to "plant" (planting his flag):

"He has planted his flag in every corner of the world"/

"His flags/banners are [firmly] planted in every corner of the world."/

"Thus he's currently planting his flag/banner around the globe."

But for its unfortunate relation with urine, you could perhaps consider "marking his territory":

"He has marked his territory in every corner of the world"/

"His territory is [clearly] marked in every corner of the world."/

"Thus he's currently marking his territory around the globe."

Strutting his stuff could work, but only if the chef is pompous/doing all this in a pompous fashion and you wish to point this out:

"Thus he's strutting his stuff in every corner of the world/around the globe"