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We describe the construction of traversable wormholes with multiple mouths in four spacetime dimensions and discuss associated quantum entanglement. Our solutions are asymptotically flat up to the presence of magnetic fluxes that extend to infinity. The construction begins with a two-mouth traversable wormhole supported by backreaction from quantum fields. Inserting a sufficiently small black hole into its throat preserves traversability between the original two mouths. This black hole can be the mouth of another wormhole connecting the original throat to a new distant region of spacetime. Our wormholes are traversable between any pairs of mouths.