If I install tin tiles over my formica backsplash will they
Stick okay. They have self-adhesive backs. My husband says they will fall off. How can I make them stay permanently.

I personally would use 100% pure silicone on the backs. Take a test piece and see how it reacts to the adhesive that's on the back of the tile.
But fist I would follow the instructions of the installation of the tile.
Many of times on the gob site I've used silicone for an adhesive the good thing is you change your mind it can be removed.
Your husband just does;t have faith in stick on glues. The other way to help the tile stick is ruff sandpaper the backsplash so it has a better surface to stick to But when you do that there is no return to original back-splash

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Clean the formica with a good solvent or degreaser. Tile companies have been getting better with their peel-n-press products so give it a chance, then if a tile does ever fall of you can then apply a thicker/stronger adhesive.