The water comes from inside the ship's bilges. Some of it is waste-water from the accommodation and engine spaces, some is sea-water and rain-water which finds its way inside the ship. No vessel is totally waterproof, especially above the waterline. Water will continually find its way inside, so ship designers build in spaces to which it is channelled called bilges. To prevent the bilges from filling up and overflowing, they are regularly, sometimes continuously, pumped out overboard. That's what this "exhaust" water is - bilge-water from the bilge-pumps
Sorry just noticed the cross-post with Cohaagen.
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