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"that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it."

Would you have some examples?

(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)

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Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]

If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.

[1] https://hsm.stackexchange.com/questions/170/how-did-group-th...

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