Base Out (BO) and Base In (BI) prism are similar, in that both types of prism bend light rays. However, the big difference is the direction they bend the light rays and why they are used.
There is no reason for the eyes to need to diverge beyond having the axes of vision be parallel for viewing distant objects. If the eyes do diverge, BI prism is used to bend the light rays toward the nose and fuse the images.
Base Out (BO) prism is used to bend light rays away from the nose and is used for several reasons. It is normal for the eyes to turn inward (converge) for close focusing. If they did not, ou would see double whenever you tried to look at something close. The three most common causes for needing BO prism are Muscle Imbalance, Cranial Nerve problems, or Eye Positioning Control Issues in the Brain. Lets tackle the Brain Issue first. There are two interconnected eye muscle control system in the brain. One system controls the ciliary muscles and the accommodation power of the crystalline lenses in your eye. The other controls the 6 muscle pairs (3 on each eye) that control the position of your two eyes.
The eye positioning system appears, from an engineering point of view, to be what is called an "Open Loop Servo" system. There are two basic types of servo positioning system. One is a "Closed Loop Servo System" where the control system knows EXACTLY where the CONTROLLED element of the system os located or pointing. These types of systems are used for precision machining where the position of the tools are known within a tiny fraction of an inch or mm. To my knowledge, there are no Closed Loop Servo Systems in the human body. However, the body is loaded with Open Loop Servo Systems where the eyes play an important role. Open Loop Servo Systems use the eyes to look at the results of movement. When you reach for an object, you almost always have to look at your hand position to be able to make nearly instantaneous position corrections. The eye positioning system uses the two images from your eyes to control the position of your eyes.
To make understanding how the system works a bit more difficult to understand, there also seems to be a two-way connection between he eye focus control system and the eye positioning system that causes the eyes to automatically converge when you try to focus close. The strength of this connection varies with individuals and is sometimes the cause of over convergence in Hyperopes (people who are far sighted) because the act of using your internal focus mechanism (accommodation) to correct your distance vision and work harder than normal to focus close, also causes the eyes to converge or cross. You can also have muscle imbalance or cranial nerve issues where the eyes want to turn inward or cross and the interconnection will sometimes cause the eyes tot try to focus close, it seems to work both ways.
The problem you and everyone else who experiences Double Vision has is that no matter what the cause, there are only three solutions. Glasses with prism correction, muscle surgery, or sometimes vision training. At some point, you will probably need to see a Pediatric Ophthalmologist (they also work with adults) who specialize in problems of this nature. However, before you do that, I would like to suggest that you do the "Simple Prism Test" described a few posts below, that I devised and posted nearly a year ago.
I need to make a few changes to the test to clarify the conditions, but I can tell you here. You need to post the calibrated strip at least 10 feet (3 meters) away so your eyes will be pretty relaxed and not inclined to converge too much because of the closeness and you need to try to let your eyes be as relaxed as possible. You may be surprised at the numbers you get, but you need to try the test when you first get up and then when you are tired. Most ECPs are trained to significantly under prescribe prism - perhaps only a 50% correction, which is not necessarily a bad thing, but it can be very annoying if your eyes break fusion too often.
You asked a question about how much priam will you ultimately need, the answer is no one really knows at this time, but it can become very significant depending on the cause. Also, you need to know that muscle surgery is not always as effective in adults as it is in young children. One thing that could be affecting an over convergence tendency is that you actually have significantly more Hyperopia than was revealed in your exam. The condition is know as Latent Hyperopia.. That is where your ciliary muscles and crystalline lenses have become so used to correcting your Hyperopia that they have trouble relaxing. The solution is to wear PLUS glasses so your eyes can gradually relax and they can eliminate the over-convergence tendency.
I urge you to perform the Simple Prism Test and get back to us with the results and we can go from there. If you don't understand anything or have any questions, please ask for clarification.
May I ask your occupation and where you live?
Sorry to be so long winded, but this is a complex subject.