Nothing in reality has an imaginary value. Measurable quantities are measurable as real numbers (along with a unit of measure to gives it real world meaning). Its possible to compute an imaginary value from real things, but that imaginary value represents a real measure of something. Pretend its real and change the unit of measure, and you have a real number again.
The fact is, no number tangibly exists. They are ALL abstractions. Some, in a practical sense, are meant to describe real things. Others, are meant to describe other abstract concepts and relationships. All of it is man-made. Numbers didnt lie around the cave floor waiting to picked up. They were invented, imagined, and have no tangibleness.
Here is an interesting thought for you to consider.
You concern yourself with imaginary numbers too much. What about real numbers?
Are negative numbers real? You can have a positive number of apples, but you cannot have a negative number of apples. You can measure a positive distance in one direction, but negative distances are impossible. Negative velocities are impossible. Rather, when we see a negative sign in distances and velocities, we change the direction of the vector. A negative distance is really just a positive distance in the opposite direction.
What about non-integers, for that matter? Anything that exists in reality can be divided into discrete units. The universe is not a continuous thing, it is a discrete thing. Plank units. Everything can be measured in whole units of something else. You can have 1 whole piece of a half-apple, but not 1/2 of an apple. You cant have half an atom. You cant have half a life. Get right down to it, on a fundamental level, we can count in whole integer discrete units. We can count atoms. Plank distances. What is a half of a mol of water? 3.01 x 10²³ atoms of water. A "dozen" is a word, a convenience of language. We can SAY we have a half-dozen. But what we really mean is we have 6. You can say you have a half of an apple but what you really mean is you have a half-apple. To say you have a half of an apple is to talk about the other half that isnt even there and to relate what is to what isnt. It makes little sense. Each half-apple is its own discrete unit and I have two of them. If I said I had two half-apples, are you immediately going to assume that the two halves came from the same whole apple? It is semantics.
Non-integers, irrational numbers, etc. and imaginary numbers, are all just phenomenon of pure, abstract mathematics. Math CAN be used to describe the real world, but it doesnt have to. And, indeed, the real world can only be described approximately, not exactly. So the reality is that it doesnt matter if integers or imaginary number or rational numbers or numbers in general are real or not. What matters is how fruitful our understanding of the world is on account of our descriptions of it.