¿could multicellular life exist without cyanobacteria?
ALMOST ALL MULTICELLULAR LIFE would NOT EXIST without CYANOBACTERIA
This question is actually a lot more nuanced than you might think and answering it took quite a bit of research! Buckle up, this is going to be a long one!
Let's start by talking about which organisms are multicellular. Although it's a common misconception that the only multicellular groups are plants, animals, and fungi, this is actually not the case! In fact, multicellularity is not just limited to cells with a nucleus/mitochondrion (eukaryotes)! There are even some multicellular bacteria. Let's look at each group of multicellular organisms and see whether they would exist without cyanobacteria!
Animals (would NOT EXIST without CYANOBACTERIA)
Animals would not exist without Cyanobacteria. I've covered this in many other posts, but in short, because all animals breathe O2 and almost all of Earth's O2 was created by Cyanobacteria, it's safe to say that all animals would NOT EXIST without Cyanobacteria.
Multicellular Fungi (would NOT EXIST without CYANOBACTERIA)
Although there are some multicellular fungi that do not breathe O2 (they're called Neocallimastigomycota. Say that 5 times fast!), all multicellular fungi still would not exist without Cyanobacteria. This is because the common ancestor of all multicellular fungi DID breathe O2, even if some of its distant grandchildren now don't. This is kind of like how you can technically say that all humans wouldn't exist without fish because the common ancestor of fish and humans was a fish. Therefore, all multicellular fungi couldn't exist without Cyanobacteria because the common ancestor of all multicellular fungi required Cyanobacteria.
Plants (would NOT EXIST without CYANOBACTERIA)
This is also something I've covered in a lot of posts, but simply put, plants would not exist without Cyanobacteria because their chloroplasts (the "organelle" in plant cells that does photosynthesis) is a Cyanobacterium. Not "evolved from a Cyanobacterium", it literally is a Cyanobacterium. Every plant that has ever existed has had Cyanobacteria inside of it, and no plant that has ever existed could survive without them.
Other multicellular eukaryotes including Red Algae, Green Algae, Stramenophiles, some Alveolates, and Choanoflagellates (would NOT EXIST without CYANOBACTERIA)
These groups of eukaryotes evolved multicellularity independently of the "main three" kingdoms of plants, animals, and fungi. However, all of these groups would NOT EXIST without Cyanobacteria for the same reason that plants wouldn't: they also have chloroplasts!
Another multicellular group of multicellular eukaryotes that would NOT EXIST without Cyanobacteria are the choanoflagellates. These are the closest relatives to the animals! Although choanoflagellates don't have chloroplasts, all choanoflagellates need to breathe O2 to survive, which we already know requires Cyanobacteria!
It is UNCLEAR whether other multicellular eukaryotes would exist without CYANOBACTERIA
We're getting into the obscure species now. Other multicellular eukaryotes including slime molds, most alveolates, and some other scattered multicellular protists might have been able to exist without Cyanobacteria, but it's unclear. Basically, the question boils down to whether the ancestors of all these different groups HAD to breathe Oxygen. Although the ancestors of all of these groups COULD breathe Oxygen, we don't know if Oxygen was a requirement or not. If O2 was a requirement, then they would not exist without Cyanobacteria. If it wasn't, then they could exist without Cyanobacteria.
SOME Multicellular Bacteria COULD EXIST without Cyanobacteria WITH ONE IMPORTANT EXCEPTION
Some multicellular bacteria require Oxygen to survive, including Myxobacteria and Actinobacteria, which means they would NOT EXIST without Cyanobacteria. Some multicellular bacteria do not seem to require Oxygen to survive, including Cable Bacteria, and thus COULD exist without Cyanobacteria. However, we as scientists don't know much about multicellular bacteria yet, so these ideas could change overtime.
However, the most important multicellular bacteria (that would not exist without Cyanobacteria) are none other than... CYANOBACTERIA! That's right! Some Cyanobacteria (not the ones in plants, but there are a LOT of different species of Cyanobacteria) are themselves multicellular!
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Even though there are some multicellular organisms that would exist without Cyanobacteria, I hope this post really goes to show just how important they are anyway!