  We present a systematic method to introduce free parameters in sets of
mutually unbiased bases. In particular, we demonstrate that any set of m real
mutually unbiased bases in dimension N>2 admits the introduction of (m-1)N/2
free parameters which cannot be absorbed by a global unitary operation. As
consequence, there are m=k+1 mutually unbiased bases in every dimension N=k^2
with k^3/2 free parameters, where k is even. We construct the maximal set of
triplets of mutually unbiased bases for two-qubits systems and triplets,
quadruplets and quintuplets of mutually unbiased bases with free parameters for
three-qubits systems. Furthermore, we study the richness of the entanglement
structure of such bases and we provide the quantum circuits required to
implement all these bases with free parameters in the laboratory. Finally, we
find the upper bound for the maximal number of real and complex mutually
unbiased bases existing in every dimension. This proof is simple, short and it
considers basic matrix algebra.
