The main goal of this paper is to construct a wavelet-type random series representation for a random field $X$, defined by a multistable stochastic integral, which generates a multifractional multistable Riemann-Liouville (mmRL) process $Y$. Such a representation provides, among other things, an efficient method of simulation of paths of $Y$. In order to obtain it, we expand in the Haar basis the integrand associated with $X$ and we use some fundamental properties of multistable stochastic integrals. Then, thanks to the Abel's summation rule and the Doob's maximal inequality for discrete submartingales, we show that this wavelet-type random series representation of $X$ is convergent in a strong sense: almost surely in some spaces of continuous functions. Also, we determine an estimate of its almost sure rate of convergence in these spaces.