When I use the tower BG, which features animated torch frames, and I switch to a different tileset, the torch animation glitches and displays as purple squares, this doesn't happen if I use the game's original tower tileset. Does anyone know why this happens and how to fix it?

did you use the original tower background (BACK) ? You need this background for the animation to work . Send The Patch / a screen shot and i might figure it out.
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This is what it looks like if I use a tileset other than the original tower one, look at the torch:



and I'm using the original tower background (back) too. (39 back and 39 front)

If you use tilset slot 1 / 32 / 58 ...(other castle duplicates)it will use the castle pallette (for the castle background)
so you need to use a different slot ,duplicate the tile set or modify the animation.
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If you see this , your probably never see It again!

i believe the grey castle tileset is loading its animation graphics over the tower bg animations. simple fix would be replacing the grey castle animations (d_2d_TEN_I_yakata_nise_ncg.bin, in the ncg folder in the file browser) with the tower animations (d_2d_TEN_I_toride_ncg.bin), but you wouldnt be able to use the grey castle background without its animations being broken.
another maybe better simple option would be to replace another tileset which doesnt have animations with a duplicate of the grey castle tileset, and use that. (as an aside, the conveyors wont animate this way, but i dont believe they animate in the original grey castle tileset anyhow)