Your intuition is good. If you are trading completely uncorrelated markets with uncorrelated signals, then one still gets a diversification benefit which bumps the portfolio Sharpe. In real life, the markets are correlated and your signals are correlated. This can help or hurt the portfolio depending upon whether you consider this in your trading.
R is a column vector of returns
w is a column vector of your portfolio weighting (w>=0) (sum w = 1)
V = covariance matrix.
R0 is Risk-free rate = 0% these days
Sharpe_port = Expected excess return / expected volatility
= R * w -R0/ sqrt( w * V * w)
If no correlation, then V is diagonal. If all N markets have equal volatility v, and an equal weighting
Sharpe_port = R/N / sqrt( v * v * 1/N) = Sharpe_each /sqrt(N)
If all of your systems on all of your markets perform equally, then the portfolio Sharpe is 1/sqrt(Number of traded market system pairs)
Adding one degree of realism, your system-market pairs are not expected to perform identically so one needs to weight intelligently. If almost all of your money is in one stock, then the portfolio is really just the one stock, and the Sharpe Ratio will follow accordingly.
Adding another degree of realism, your system-market pairs are not uncorrelated. This is especially true when major adverse affect occur. If you want to buy stocks that all fit the same mold, then bad news affects several trades so w * V * w is closer to N*N*v*v/N/N = v*v so you expect the average return with no diversification benefit.
One can keep at this, and do some really sophisticated money-management/portfolio allocation math to look at your current positions and your current opportunities to find your smartest moves. For example, if you already have a long Euro trade, and have a signal for a short Swiss Franc trade, should you resize your Euro trade? Good traders know how their markets are related, and take this kind of thing into account. Objectively doing this has a bad name. Real-life rebalancing costs are substantial, and too often these models are unrealistic. I have my own way of doing this.
One easy way to get some diversification benefit without going huge on analysis is to use pair-trading/spread trading. If you want to bet on the general market, then one can just bet an index. If you have an opinion on a particular stock or future, then find a correlated/cointegrated reference point. For example, maybe you like a particular gold mining stock relative to an index of gold mining stocks. If you are right, then buying one and selling the other should net a positive return, but if balanced, a low volatility. This makes for a good Sharpe Ratio.
By making a portfolio of pair-trades, one can lower volatility through diversification. (How correlated is a gold stock vs a gold index trade when compared with an Internet stock vs a technology index?)
By minimizing the volatility, one can trade larger for the same risk, and make more money off of the same trading opportunities.