I have a tutoring business. If I pay my spouse to work as a registrar and materials coordinator, do I have to issue a Form 1099?

You’re not asking the right question. You should be asking whether your spouse is your employee. Under the circumstances described, where your spouse is working solely for you and you have control over the work being done, it is likely that your spouse is your employee rather than an independent contractor. In this case, you must adhere to all employer obligations, including tax withholding, paying payroll taxes, and issuing a Form W-2.