I am wanting to replace this bathtub faucet without changing the plumbing behind the wall. I am assuming by identifying the brand/type of faucet already installed I could have success by getting something similar?
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Unfortunately, plumbing fixtures have no constancy. There are many, many different makes, and no company makes the same thing one year to another. They all change their models every year, so finding the same fixture is virtually impossible. That said, the stuff behind the fixture tends to be standardized. The handles are attached to what are called "faucet posts". This is what they look like:
Most ordinary turn handles will work, and if not, hey, just return them for store credit.