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After swimming in the ocean, I noticed a briny taste on my lips from the salty water.
Saltwater is briny, so any ocean-going vessel is said to "sail the deep and briny".
The Beautiful Briny was created in 1971.
Briny refers to salty water so on the briny would refer to being on a body of salty water.
Interpreting the expression 'on the briny' as 'on the sea', the appropriate pronoun is it: 'on it'.