The Coriolis force is not a real force, but depending on the reference system and how it does refer to an inertial system. Could this Coriolis force as widely used in atmospheric/climate modeling be replaced by an explicit force eg from sun gravitation and would this give the same results?
Here are your choices with regard to modeling the atmosphere. There aren't many, and only one of them makes sense.
Model the atmosphere from the perspective of an inertial frame of reference.
Good luck with that! As an advisor told me decades ago, "Name one!" It's certainly not an Earth-centered frame; the Earth is orbiting the Sun. It's certainly not Sun-centered frame; the Sun is orbiting Jupiter, Saturn, Uranus, and Neptune. It's certainly not the solar system barycenter (center of mass); the solar system is orbiting the Milky Way. The Milky Way barycenter? Nope. The Milky Way and the Andromeda galaxy are accelerating toward one another. No matter how far afield we look, we see gravity (and even further afield, metric expansion of space) that hides this inertial frame from us.
Use a non-rotating frame centered at the solar system barycenter.
The International Celestial Reference Frame is one such frame. This is more or less a constructible frame. Here we either ignore or account for the acceleration toward the center of the Milky Way, the Andromeda galaxy, and stuff further afield. It does makes sense to ignore those extraneous accelerations; they are very, very tiny, and the variation across the solar system is tinier yet. There's another issue with any such construction. The ICRF almost certainly is rotating. The predecessor of the ICRF, the J2000 frame: Scientists now know it was rotating. The same goes for the predecessor of the J2000 frame, the B1950 frame. In fact, scientists know the ICRF is rotating. The ICRF has been upgraded to ICRF version 2.0. And that too is rotating. Scientists just don't know how much.
Use a non-rotating frame centered at the Earth's center of mass (aka Earth-centered inertial).
Even if the ICRF was perfect, there is no way one can reasonably model the Earth's atmosphere in a computer program from the perspective of such a frame. The length and velocity scales needed to properly model the atmosphere and the length and velocity scales using a solar system barycenter frame just don't match. Oil and water. Even worse, it's a Sesame Street moment! ("One of these things is not like the others, One of these things just doesn't belong.")
The thing that doesn't belong is the solar system barycenter. The only reasonable way to model the Earth's atmosphere is to use an Earth-centered perspective. Now a computer program is starting to have a chance to represent the atmosphere, and do so in a way that is computationally feasible. An Earth-centered frame is accelerating, but we can handle that. We can pretend that $\vec F=m \vec a$ still applies. This result in fictitious tidal accelerations. It's not a problem. Physics still works.
Use an Earth-centered, Earth-fixed reference frame.
A problem remains with the Earth-centered inertial frame described in option #3. The Earth, the atmosphere, and the oceans are rotating, and at different rates. This means that "starting to have a chance" was a bit overoptimistic. It makes much, much more sense to use a frame that rotates with the Earth. This adds even more fictitious forces, the centrifugal and Coriolis forces, but once again, that's not a problem. In fact, this simplifies things to the point where meteorologists do finally have a chance to represent the atmosphere in a way that is computationally feasible.
Option #4, the one that requires fictitious third body forces (aka tidal forces), fictitious centrifugal forces, and fictitious Coriolis forces, is the only option that makes a bit of sense from the perspective of modeling the atmosphere.
The Coriolis acceleration is only present in a rotating reference frame as is the case with Earth. The Coriolis effect is caused by Earth's rotation and the inertia of the mass experiencing the effect. If you are in an inertial frame of reference, thus non-accelerating, there will be no Coriolis effect.
Let's assume that you are capable of modeling the motions on Earth as part of a Solar System model with the frame of reference centered at the Sun (and assume that it is non-accelerating), then there will be no need to include the Coriolis effect.
If your question is: I have an equation with force term $F(x,t)$, and suppose that $F(x,t)$ is caused by effect A, then will the solution of the equation be the same as if the force $F(x,t)$ had been caused by effect B, then the answer is of course "yes". Same force (i.e., same magnitude, same direction) will always cause the same reaction of the system, regardless of where this force comes from.
Of course, the coriolis force has a very specific form and you will have difficulty finding any other physical effect that would lead to a force term that happens to have the exact same form.