We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is Zn, this bound is 2n +1, while the Khovanskii bound is exponential in n2. The bound 2n + 1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.