Since the speed of all motions are limited, the time-delay phenomena are often inevitable in the signal transmission or other process. The fractional functional differential equations are important models to study the movement of time-delay systems. When there are two or more interact state variables in the system, they could always be characterized by coupled differential equations. The existence and uniqueness of positive solutions for boundary value problems of a class of nonlinear delay coupled functional differential systems with Riemann-Liouville fractional derivatives are studied. Firstly, according to the characteristics of equations and boundary conditions, a comparison theorem for the system is constructed, the monotonic sequence of upper and lower solutions is obtained, and the relationship between the upper and lower solutions is determined. Secondly, the existence theorems for positive solution of boundary value problem are established and proved by using the method of upper and lower solutions, and the value range of positive solutions is obtained. And thirdly, the existence and uniqueness theorem for positive solution of the boundary value problem is established and proved by iterative technique. Finally, a specific example is given out to illustrate the adaptability and universality of the main results.