  In recent years, many positivity-preserving schemes for initial value
problems have been constructed by modifying a Runge--Kutta (RK) method by
weighting the right-hand side of the system of differential equations with
solution-dependent factors. These include the classes of modified
Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods.
Compared to traditional RK methods, the analysis of accuracy and stability of
these methods is more complicated. In this work, we provide a comprehensive and
unifying theory of order conditions for such RK-like methods, which differ from
original RK schemes in that their coefficients are solution-dependent. The
resulting order conditions are themselves solution-dependent and obtained using
the theory of NB-series, and thus, can easily be read off from labeled N-trees.
We present for the first time order conditions for MPRK and GeCo schemes of
arbitrary order; For MPRK schemes, the order conditions are given implicitly in
terms of the stages. From these results, we recover as particular cases all
known order conditions from the literature for first- and second-order GeCo as
well as first-, second- and third-order MPRK methods. Additionally, we derive
sufficient and necessary conditions in an explicit form for 3rd and 4th order
GeCo schemes as well as 4th order MPRK methods. We also present a new 4th order
MPRK method within this framework and numerically confirm its convergence rate.
