  For supersymmetric gauge theories with eight supercharges in four, five and
six dimensions, a conserved current belongs to the linear multiplet. In the
case of six-dimensional $\cal N=(1,0)$ Poincar\'e supersymmetry, we present a
consistent deformation of the linear multiplet which describes chiral
anomalies. This is achieved by developing a superform formulation for the
deformed linear multiplet. In the abelian case, we compute a nonlocal effective
action generating the gauge anomaly.
