'Time is not real' should be understood not as the claim that you do not experience time, but rather time is constructed by our brains and is not strictly speaking physical.
From the SEP article Time:
In a famous paper published in 1908, J.M.E. McTaggart argued that there is in fact no such thing as time, and that the appearance of a temporal order to the world is a mere appearance. Other philosophers before and since (including, especially, F.H. Bradley) have argued for the same conclusion.
Going back to the Hobbesian notion that change is motion (PhilSE), and as the user SystemTheory has noted, time is a shorthand for motion. A year is the motion of the earth about the sun. A day is the shorthand for the motion of the earth spinning relative to the sun. An hour is 24th of a day, and so on. Consider what seems so revolutionary when you first learn it, that time is relative to a frame of reference, really isn't. Time dilation is non-Netwonian because of the metaphysics that says the speed of light impacts our experience of clocks. You cannot smell time. You cannot feel time. You cannot weigh time or put it in a container. You can simply experience it by noting change. 'Time' in the abstract, simply is experience generalized over concrete experiences of definitive periods of time. The next best definition are the circular definitions of time, velocity, and motion, each defined in terms of the other.
Dennett is his book Consciousness Explained has a chapter devoted to the psychology of time which is much more contemporaneous and scientific than Kantian language. On page 144, in the chapter Time and Experience, Section 2. How the Brain Represents Time, he goes out to map how the brain functions in architectural terms and makes a simple argument. Time is a type of non-sensory neural calculation that is used to help sequence bodily events. As such, the building blocks of consciousness, events, are nothing more than a collection of events which are themselves delimited by two unique events: one of which marks the beginning and one which marks the end creatively delimiting events recursively temporal content.
Time as a construct of the mind is a concept supported by the framework of embodied cognition (SEP). The basic premise of embodied cognition is that the products of the mind, that is mentality and to a wider degree philosophical intentionality are directly grounded in, shaped, and constrained by the nature of the body that produces the mind. (Yes, there's a metaphysical presumption about the relationship between mind and brain.) In fact, it is a central contention of embodied cognition that mind is understood much more broadly than mere brain activity, but includes the body itself supported by such hypotheses as the somatic marker hypothesis and provides a foundation that cognition itself might be understood as extended or social in nature.
Given that we are born with naive realism as our fundamental cognitive strategy for organizing our mental activities, it is important to reflect on 'real' as understood through the lens of broader meta-ontological commitments. If your notion of 'real' is tied up in empirical claims, then it might sound nonsensical to declare time imaginary if imaginary is used as a synonym for bogus, illusory, or not accessible to the public. You invoke the term reality, but to a conceptualist like McDowell, it is important to differentiate among physical, perceptual, conceptual, and social forms of reality.
Classic physics notions like time, space, and energy aren't sensible. They are instead intuitive faculties that allow our consciousness to emerge and persist. They are not directly available to our senses, but rather allow our senses to come into being. This is what makes Kant's project of marrying Newton to Leibniz so fruitful. The recognition that time like space are intuitions that inhere to the mind helps us to understand the role intuition plays in framing our conscious experience and reframing the conversation in a layered model of reality leading from the noumenological with its pervading ontological ambiguity through the mysteries of qualia right to understanding the role language plays in shaping our concepts.
What about 'real time'? Doesn't time have to exist external to our thoughts? Well, that's not so clear. When talking about any form of reality, we are dealing with our thoughts about what reality is. If I make the claim 'time is real', reality itself is our model of the external world, and that model or any model of the universe is itself inescapably grounded in thought. This may be why the Wikipedia article on time says:
Defining time in a manner applicable to all fields without circularity has consistently eluded scholars.
Is human-like intelligence predicated upon this basic architectural feature of the embodied mind to create temporal content to augment mental representations for the purpose of sequencing and coordinating among the disparate parts of the body and keeping the sensory percepts organized? I would speculate yes. In much the same way that computer architecture has to have common features to embody physical computation, so too I suspect does language use and logical consequence depend fully on the same sorts of bodies that we broadly call animals.
For more information, see:
The Experience and Perception of Time (SEP)