We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of C∗-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group G. A continuous Tambara-Yamagami tensor category associated to a locally compact group G is a continuous tensor category that has a single non-invertible simple object τ such that τ⊗τ
Nearby in the stack
decomposes as a direct integral indexed over
G
, meaning
τ⊗τ≅L2(G)
. We show that continuous Tambara-Yamagami tensor categories for
G
are classified by a continuous symmetric nondegenerate bicharacter
χ:G×G→U(1)
and a sign
ξ∈{±1}
. We also prove that, if a
W∗
-tensor category
C
obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that
C
is obtained from a continuous tensor category by forgetting its topology.