The following leads to extremely long execution times:
using BlockArrays
A = BlockedArray(rand(ComplexF64, 5000, 5000), [2000,2000,1000], [2000,2000,1000])
B = BlockedArray(rand(Float64, 5000, 2000), [2000,2000,1000], [2000])
C = complex(B)
@time A * B
@time A * C
The latter finishes in 1.7 seconds, whereas the first (mixed realness) takes more than 100 times longer.