We investigate a signed version of the Hammersley process, a discrete process on words related to a property of integer sequences called heapability (Byers et al., ANALCO 2011). The specific version that we investigate corresponds to a version of this property for signed sequences. We give a characterization of the words that can appear as images the signed Hammersley process. In particular we show that the language of such words is the intersection of two deterministic one-counter languages.