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When starting the Tidal Basin mission, there is usually no background music. I've found that exiting the mission, starting another mission, then leaving that one and returning to Tidal Basin usually results in the background music being present. However, this hasn't worked with every mission; switching to Jefferson Plaza usually works for me. This is shown in the linked video. I haven't had issues with any other mission not having background music. I should also note that I haven't had this issue when playing the Tidal Basin Manhunt mission where Hornet is the prime target.
This is a long-standing issue. The first two times I played through Tidal Basin (at the end of World Tier 4 and during World Tier 5), I assumed the lack of background music was a deliberate design choice, and I was surprised to hear music when I played the mission again later.
While I'm at it, I'll add that the description of the mission ("Breach the Black Tusk Compound and Prevent Missile Launch") really should be changed. Aside from the capitalization and grammar issues, it spoils the climax of the mission for new players. You don't even know about the threat of a missile attack until you're near the end of the mission, nor is that your objective when you start. It should be more like "Breach the Black Tusk compound and retrieve the broad spectrum antiviral before it can be extracted from the city." This is a more accurate, less lazy-sounding description.
