We present a new class of on-demand routing protocols called Split Label Routing (SLR). The protocols guarantee loop-freedom at every instant by ensuring that node labels are always in topological order, and thus induce a directed acyclic graph (DAG). The novel feature of SLR is that it uses a dense ordinal set with a strict partial order to label nodes. For any two labels there is always some label in between them. This allows SLR to "insert" a node in to an existing DAG, without the need to relabel predecessors. SLR inherently provides multiple paths to destinations. We present a practical, finitely dense implementation that uses a destination-controlled sequence number. The sequence number functions as a reset to node ordering when no more label splits are possible. The sequence number is changed only by the destination. Simulations show that our proposed protocol outperforms existing state-of-the-art on-demand routing protocols.