  We investigate the Calder\'on problem for the fractional Schr\"odinger
equation with drift, proving that the unknown drift and potential in a bounded
domain can be determined simultaneously and uniquely by an infinite number of
exterior measurements. In particular, in contrast to its local analogue, this
nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness
result is complemented by an associated logarithmic stability estimate under
suitable apriori assumptions. Also uniqueness under finitely many
\emph{generic} measurements is discussed. Here the genericity is obtained
through \emph{singularity theory} which might also be interesting in the
context of hybrid inverse problems. Combined with the results from
\cite{GRSU18}, this yields a finite measurements constructive reconstruction
algorithm for the fractional Calder\'on problem with drift. The inverse problem
is formulated as a partial data type nonlocal problem and it is considered in
any dimension $n\geq 1$.
