To begin, I would like to point out the faulty labeling of the two sides of this argument. As it is stated, it implies that believing in the relativity of time denotes a disbelief in time being real. A better argument would be between time being real or an illusion, consistent or relative, as one could believe time to be a consistent illusion or a real relativity. I am of the latter school of thought, but I've chosen to write under Relative, supposing that Real is meant to imply that time is consistent.
Time is often argued to be arbitrary, due to the fact that one day, defined as the time it takes a planet to complete one rotation, will span different lengths of actual time depending on the planet. A day on Saturn is less than half the length of a day on Earth. However, this does not mean that time is arbitrary, but rather our system of measuring it. Due to a lack of necessity, we have not devised a universal system for measuring time. If we colonize other planets and keep regular communication among them, such a system will undoubtedly be implemented. For now, we simply state everything in terms relative to Earth time.
Therefore, a system of labeling time is arbitrary, but time itself does exist. For one, wormholes are theorized to connect points in time in addition to space. Furthermore, the development of a theory of everything hinges in part on regarding time as one of the dimensions. This means that there is a universal flow of time, indicated by its ability to be scientifically categorized as a dimension and consist of points. The trouble lies in developing a universal scale, as few populations of similar objects all behave exactly the same.
Just as other physical dimensions and measurements can be manipulated, though, so too can time. Heat can increase volume and thus decrease density. Pressure can decrease volume and thus increase density. Pressure as a result of positive velocity also decreases mass. And positive velocity also decreases time, or rather the movement of it.
This is one point where regarding time as a dimension becomes tricky. It can refer to either a span of time, such as one hour, or the rate of passage, such as 364.9999999683 days per year (the rate of time for astronauts in orbit). Regarding it as akin to length, there does in fact exist a universal span, thus far not manipulable. Suppose we were to use the half-life of an atom of hydrogen as a universal unit of time, and we decide to call it a clack. In our current basis of time, on Earth and at rest, 12.32 years would pass in that time. On Saturn and at rest, approximately 125.7 years would pass. For a being zipping through space in a shuttle, a fraction of that time would pass, the actual number depending on what scale they use. But, just as 1000 pounds or 71.43 stones is still half a ton, 12.32 Earth-years or 125.7 Saturn-years would equal 1 clack.
Regarding time as akin to velocity, that is as a rate of change, it becomes more complex. As was mentioned in the two previous paragraphs, time as a rate is inversely relative to velocity. The faster we move, disregarding the motion of the planet, the slower time moves. But, returning to the previous example of Earth-years and clacks, while we may be able to age only 12 Earth-years over the course of 12.32 Earth-years, 1 clack would still have passed throughout the universe. However, this does prove that the rate of time is manipulable and thus relative, even if the span is not, despite the absence of a universal scale.
If we argue that this makes time arbitrary, the fact that we can manipulate our own person flow of time, then we can argue that chemistry calls into question all atomic measurements and thus makes all of them arbitrary. Really, it's just the fact that our label and scale for time is inconvenient on other planets. But since heat and pressure manipulate volume and mass, this means that those measurements are inconsistent and thus inconvenient between Earth and most other planets.
Still, in spite of this, the fact that spans of time are just as valid a universal measurement as any other regardless of the scale currently used, the flow of time is very much relative. Developing a universal for measuring spans will spawn one for rates of time, such as 12.32 years/clack on Earth and 125.7 years/clack on Saturn. Until then, we have to make do with stating other planets' times in terms relative to Earth's, such as 10.2 Earth-years/Saturn-year. In the case of objects moving at a high velocity (any velocity will reduce the rate of time, but it's barely measurable at such velocities as we reach while driving), the rate is determined by multiplying a unitless time dilation factor by a given at-rest base.