We give a time-randomness tradeoff for the quasi-random rumor spreading protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete graphs. In this protocol, the goal is to spread a piece of information originating from one vertex throughout the network. Each vertex is assumed to have a (cyclic) list of its neighbors. Once a vertex is informed by one of its neighbors, it chooses a position in its list uniformly at random and then informs its neighbors starting from that position and proceeding in order of the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009] showed that after $(1+o(1))(\log_2 n + \ln n)$ rounds, the rumor will have been broadcasted to all nodes with probability $1 - o(1)$. We study the broadcast time when the amount of randomness available at each node is reduced in natural way. In particular, we prove that if each node can only make its initial random selection from every $\ell$-th node on its list, then there exists lists such that $(1-\varepsilon) (\log_2 n + \ln n - \log_2 \ell - \ln \ell)+\ell-1$ steps are needed to inform every vertex with probability at least $1-O\bigl(\exp\bigl(-\frac{n^\varepsilon}{2\ln n}\bigr)\bigr)$. This shows that a further reduction of the amount of randomness used in a simple quasi-random protocol comes at a loss of efficiency.