Spectral analysis of positive signals
To give a complete answer to this question you're going to need to provide more details about the kind of models you're considering in the first place. But yes, in many cases you can augment those models with a priori constraints on $Y[t]$, such as $0 \leq Y[t] \leq 1$. 

For example, if the standard model has some sort of least-squares structure, then adding constraints of that type turns the problem into a bound-constrained least squares problem. There are a variety of approaches to solving such problems, and while they are more expensive than standard least squares, they are quite tractable. And it's very likely that such constraints will produce a better reconstruction.

Even without knowing more, though, I will say this: if your modeling approach does not produce real signals naturally, then you are almost certainly using the wrong modeling approach. It concerns me that you are even proposing taking the real part of the output of some other model. You should be searching the space of real signals if you know that to be the underlying structure.