Using exceptional field theory we construct supersymmetric warped AdS 7 vacua of massive IIA and AdS 6 vacua of IIB, as well as their consistent truncations including vector multiplets. We show that there are no consistent truncations of massive IIA supergravity around its supersymmetric AdS 7 vacua with vector multiplets when the Romans mass is non-vanishing. For AdS 6 vacua of IIB supergravity, we find that in addition to the consistent truncation to pure F(4) gauged SUGRA, the only other half-maximal truncations that are consistent result in F(4) gauged SUGRA coupled to one or two Abelian vector multiplets, to three non-Abelian vector multiplets, leading to an ISO(3) gauged SUGRA, or to three non-Abelian plus one Abelian vector multiplet, leading to an ISO(3) × U(1) gauged SUGRA. These consistent truncations with vector multiplets exist when the two holomorphic functions that define the AdS 6 vacua satisfy certain differential conditions which we derive. We use these to deduce that no globally regular AdS 6 solutions admit a consistent truncation to F(4) gauged SUGRA with two vector multiplets, and show that the Abelian T-dual of the Brandhuber-Oz vacuum allows a consistent truncation to F(4) gauged SUGRA with a single vector multiplet. © 2019, The Author(s).