A Russian language exchange partner of mine boasted that unlike words in the English language, Russian words are spelt and pronounced basically the same. In response, I asked him about Арбатская and Орбацкая, pointing out that these so differently looking words must have exactly the same pronunciation. He got confused and said that Орбацкая is a non-existing word. Yet, typing it in VK, a Russian social network, I found a Russian girl with that surname, Орбацкая, and showed it to my language exchange partner. He then mumbled that I wasn't quite right about Арбатская and Орбацкая being pronounced exactly the same, and changed the topic. For the sake of politeness, I didn't press him to elaborate, but I'm curious.
Are Арбатская and Орбацкая pronounced exactly the same? If they aren't, what's the difference? Let's suppose Даша Орбацкая calls an hotel and says in Russian, "Hi, I want to book a room, my name is Даша Орбацкая." Hearing this, will the receptionist write down the surname correctly?