Topological couplings play a prominent role in our understanding of field theories, but their role in string compactifications is much less understood. I explore an example in compactifications of type IIB string theory with sixteen supercharges, showing that they lead to previously unknown components of moduli space in nine, eight, and seven dimensions. I also explain why a similar construction fails to produce a new string theory in ten dimensions. The new components of moduli space thus constructed feature an incomplete lattice of BPS strings, which has implications for a number of Swampland constraints.