As far as I understand, one can make a monoid from the space of vector bundles on a (compact) manifold $M$, with respect to the direct sum operation $\oplus$. In order to make this into a group, one needs to add 'inverses' and this leads to the consideration of formal differences $V-W$ of vector bundles. One can again compute Euler classes, Chern classes, etc... for these objects called 'virtual bundles'.

Now my question is:

When is such an object an actual vector bundle? Is there an easy criterion to check? For instance, if I have a complex virtual vector bundle and I know that its Chern classes are integral i.e. in $H^*(M,\mathbb{Z})$, rather than $H^*(M,\mathbb{Q})$, is that sufficient?

Disclaimer: I am only familiar with basic aspects of $K$-theory, so the above question might be completely trivial, but I couldn't find an answer.