Wyoming Entity is Separate
Your Wyoming corporation is a separate entity from your home state corporation. A corporation is an entity created by law, separate from you or any other corporation.
Whenever your home state corporation writes a check for valid business expenses to your Wyoming corporation, you have created a tax deduction in your home state. The income has been transferred to tax-free Wyoming.
Set up your special tax-free benefits, such as a medical reimbursement plan, retirement plan, etc., with the Wyoming corporation (which will have no employees). This will enable you to take advantage of many legal ways to take money out of your corporation tax-free, without passing those same benefits on to other employees. (Some plans, such as a medical reimbursement plan, must be available to all employees.)