The error in your numerical example is in the LHS: without rebalancing, after period 1, the weights of the assets are not 50% each, but 46% and 54%. So the portfolio return in period 2 is not 50%, but 49.3%.
Addition in response to the comment
Without rebalancing: to get the portfolio return up to some point in time, multiply the total return of each asset up to this point by its initial weight and sum all terms.
With rebalancing: multiply each asset's return in a specific period with its initial weight for that period; then summing across assets will give you the portfolio return for that period. The weights for that period can, in principle, be arbitrary: the assumption is that you rebalance the portfolio at the close of the previous period to achieve these weights. (Which may be unrealistic/impractical: just think of daily data; which is why buy-and-hold returns are often preferred to evaluate strategies). Finally, chain together these period returns to arrive at the total return for the portfolio.