So I asked on reddit, and got this answer from Rasseren :
The integration order of the endogenous variables is most often used to ensure reasonable stability (all eigenvalues of the companion form <1 in modulus) of the process; remember stability implies covariance stationarity in theory.
There is nothing, except maybe the underlying theoretical model e.g. Taylor-rule, prohibiting differing integration orders. In fact, most high-profile VAR studies utilizes different integration orders, see e.g. Stock & Watson (JEP, 2001), Bernanke et al. (Quarterly Journal of Economics, 2005) even the seminal studies of Christopher Sims (1972,1980).
The difference lies only in the interpretation of coefficients and IRFs (growth rates vs. level effects).