Waterproof doors for bathroom cupboard

I'm currently redoing my bathroom.
Right next to the bath (which also has a power shower going into it) is a heating cupboard. Previously there was an additional shower curtain used to cover the cupboard double doors and keep water from splashing onto them, but now I'd like a better solution.
I am tiling the whole bathroom, so I am toying with idea of tiling two small sheets of ply to make seamless/invisible type cupboard doors. I would also need some kind of white rubber (which would look like the grout) round the edges and frame to seal the doors watertight. The doors will be opening out the way (I thought about sliding doors but they are not really feasible because of pipes).
What would be the best way of going about this? Doesn't matter too much if it doesn't look perfect, my main concern is the waterproofing.
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