Using mesoscopic interfacial models and microscopic density functional theory we study fluid adsorption at a dry wall decorated with three completely wet stripes of width L separated by distances D_{1} and D_{2}. The stripes interact with the fluid with long-range forces inducing a large finite-size contribution to the surface free energy. We show that this nonextensive free-energy contribution scales with lnL and drives different types of bridging transition corresponding to the merging of liquid drops adsorbed at neighboring wetting stripes when the separation between them is molecularly small. We determine the surface phase diagram and show that this exhibits two triple points, where isolated drops, double drops, and triple drops coexist. For the symmetric case, D_{1}=D_{2}≡D, our results also confirm that the equilibrium droplet configuration always has the symmetry of the substrate corresponding to either three isolated drops when D is large or a single triple drop when D is small; however, symmetry-broken configurations do occur in a metastable part of the phase diagram which lies very close to the equilibrium-bridging phase boundary. Implications for phase transitions on other types of patterned surface are considered.