Should you unify colour with the branding?
Yes, the menu and the poster use the colour already on the sign or in the room, not a new palette for each piece. Do not unify around a colour the print paper cannot hold.
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Bring a daytime photo of the sign and, if you have it, the colour name the sign-maker used. The menu debuts a blue that is not on the front and they look like two venues. The colour reference written on the print sheet and applied to menu and poster.