You can get tile refinished or even refinish it yourself. I haven't done it, but I would imagine it works better on vertical surfaces than floor for the do-it-yourself kind. My grandparents had their tile re-glazed along with their ugly sink and tub, and it looks like it came in white originally. I am not sure of the cost. My other suggestion is to go bold and and mod and make it look like the retro is intentional.

This shows some bold use of blue, which is opposite the color wheel of orange. Bring in a mod light fixture and some art to tie it all together.

This is an example of heavy bold tile in a bathroom with monochromatic paint scheme.

The third picture shows an orange tile bathroom with mod fixtures. If you don't want to refinish the tile or replace it, but want real impact, I wold put in some vessel sinks and replace the shower doors while doing something bold with color on the walls.

You could put beadboard paneling over the tile walls and the tub facing. That's relatively inexpensive and would hide a lot. Then paint your window molding white to tie in with the beadboard.

I agree.. you could definitely refinish the tile and buy a refinishing kit and DIY. But if you don't want to do that at all, then I would suggest getting some thick crown moulding trim around the window. Redo the closet doors with something more contemporary maybe and make the mirror more decorative (again with maybe some moulding or tile.) I would also reframe the art to match your new sinks.
We have a selection of closet door with the look you may want or need. I've attached some images for you.
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