  We study the problems of consistency and of the existence of
finite-dimensional realizations for multi-curve interest rate models of
Heath-Jarrow-Morton type, generalizing the geometric approach developed by T.
Bj\"ork and co-authors in the classical single-curve setting. We characterize
when a multi-curve interest rate model is consistent with a given parameterized
family of forward curves and spreads and when a model can be realized by a
finite-dimensional state process. We illustrate the general theory in a number
of model classes and examples, providing explicit constructions of
finite-dimensional realizations. Based on these theoretical results, we perform
the calibration of a three-curve Hull-White model to market data and analyse
the stability of the estimated parameters.
