We derive an achievability bound to quantify the performance of a type-based unsourced multiple access system -- an information-theoretic model for grant-free multiple access with correlated messages. The bound extends available achievability results for the per-user error probability in the unsourced multiple access framework, where, different from our setup, message collisions are treated as errors. Specifically, we provide an upper bound on the total variation distance between the type (i.e., the empirical probability mass function) of the transmitted messages and its estimate over a Gaussian multiple access channel. Through numerical simulations, we illustrate that our bound can be used to determine the message type that is less efficient to transmit, because more difficult to detect. We finally show that a practical scheme for type estimation, based on coded compressed sensing with approximate message passing, operates approximately 3 dB away from the bound, for the parameters considered in the paper.