Can Big Brother decide what is real and what is not?

The method by which a totalitarian society such as that depicted in 1984 operates is essentially to try to destroy the concept of objective reality. "Truth" is made into not what people's own senses tell them—what they factually observe and remember—but instead what the Party tells them it is. The most obvious case in which this is shown to Winston is when, during his "re-education" process, O'Brien tortures him into "believing" the number of fingers he is holding up is not what Winston sees, but whatever the Party tells him to see. Earlier in the story, the Oceanians are suddenly told that they are at war with Eastasia and in alliance with Eurasia and have always been—the exact opposite of the truth—and yet everyone is expected to believe this as if nothing out of the ordinary has happened. Thus, Big Brother, as the face of the regime, is dictating "reality" to the population. It is absurd, of course, but Orwell's depiction is merely an exaggerated form of what the Communist and Fascist regimes of the twentieth century were doing in manipulating their own people—feeding them "alternative facts" in contradiction to the actual reality observed by people.

Absolutely -- this is George Orwell's scary thesis of 1984. Big Brother warps everyone's sense of reality through indoctrination programs (like the programmed displays of "hate"), historical manipulation (like Winston's responsibility to discard information that portrays the Party unfavorably), over-stimulation, and propaganda. When introduced to this world, the reader might ask themselves: "Can't someone in this world know the truth? Doesn't everyone have the capacity to know it deep inside themselves?"
This is the question that drives Winston Smith through the story. The Party decides that "2 + 2 = 5," where very basic instinctual logic dictates that 2 + 2 =4. You could say that Winston's desire is to bring back a society where 2 + 2 = 4, where the truth can be openly admitted and appreciated. O'Brien, a character supposedly belonging to a revolutionary group called the Brotherhood, inspires Winston to take action. Winston meets with him, hoping they will form a rebellion. When it turns out that O'Brien is actually a member of the Party, Winston Smith is tortured and told to admit that "2 + 2 = 5." Eventually, he does. 
So, back to the original question: can Big Brother control what is real?
The answer is yes and no. If someone lives under Big Brother, they may have intuitions and a sense of the truth, but Big Brother will find out. When he does, they will inevitably submit.
 
  

In 1984, Big Brother can absolutely decide what is real and what is not. In fact, to do this, all Big Brother and the Party need to do is to control the flow of information, and as we see from the story, that is constantly happening in Oceania.
Let's take Winston's job for example. It is his responsibility to rewrite the past according to the Party's instructions. When he is told to write about an upcoming chocolate ration, for example, Winston is rewriting the past and changing what is real. It doesn't matter that the Party said there wouldn't be a chocolate ration; that phrase has been erased from history and is no longer real, because it doesn't exist.
Similarly, when it comes to the war in Oceania, the Party switches from having a war with Eurasia to Eastasia. As Winston says in part 2, chapter 9, the Party does not admit that a change has taken place. It doesn't have to because nobody would dare to argue otherwise. What this shows is that the Party is able to determine what is real and what is not, without explanation, because people are too afraid to disagree. They simply accept what they are told as fact.