  The real Fourier-Mukai (RFM) transform relates calibrated graphs to so-called
"deformed instantons" on Hermitian line bundles. We show that under the RFM
transform, complex Lagrangian graphs in $R^{2n} \times T^{2n}$ correspond to
Sp($n$)-instantons over $R^{2n} \times (T^{2n})^*$. In other words, the
deformed Sp($n$)-instanton equation coincides with the usual Sp($n$)-instanton
equation.
  Motivated by this observation, we study Sp($n$)-instantons on hyperkahler
manifolds $X^{4n}$, with an emphasis on conical singularities. First, when $X =
C(M)$ is a hyperkahler cone, we relate Sp($n$)-instantons on $X$ to tri-contact
instantons on the 3-Sasakian link $M$ and consider various dimensional
reductions. Second, when $X$ is an asymptotically conical (AC) hyperkahler
manifold of rate $\nu \leq -\frac{2}{3}(2n+1)$, we prove a Lewis-type theorem
to the following effect: If the set of AC Sp($n$)-instantons is non-empty, then
every AC Hermitian Yang-Mills connection over $X$ with sufficiently fast decay
at infinity is an Sp($n$)-instanton.
