Is the mid-market currency exchange rate something real? Or are bid and ask rates the real rates? Reading from this and other sources, it appears to me that there are three different ratios between any two currencies, namely


the bid rate
the ask rate
the real rate


My understaning is that the two primal ratios are 1 and 2, while 3 is just a derived quantity, specifically the mid-point between the former two. In other word, I understand that 1 and 2 are the two real, market-driven numbers, and that 3 is just a convenience to know where 1 and 2 roam in that specific historical moment.

Is my understanding correct?

In the following I just report my line of reasoning.

For instance, if I have an amount $A_1$ in one currency and I want to exchange it with another currency, I would receive the amount $A_2$ in this other currency, defined as

$ A_2 = E_{1 \rightarrow 2} A_1$

where $E_{1 \rightarrow 2}$ is the exchange rate.

If I change my mind and decide to convert $A_2$ back to the previous currency, then I will be given the amount $\overline{A_1}$,

$ \overline{A_1} = E_{2 \rightarrow 1} A_2 = E_{2 \rightarrow 1} E_{1 \rightarrow 2} A_1$

I've understood that, if we leave the time elapsed between the two operations aside (or if we do the two exchanges at exactly the same moment), we are sure that $\overline{A_1} &lt; A_1$, because we are sure that in any given moment in time the following holds: 

$E_{2 \rightarrow 1} &lt; \displaystyle\frac{1}{E_{1 \rightarrow 2}} \tag{1}$

Is this true?

To me it seems obvious, since if $(1)$ had an equal sign, that would mean that I can freely exchange between currencies at zero cost (hence I could lose or earn money only as a consequence of the change of that unique rate over time due to other factors).
Indeed the bid and ask spread is very real. The liquidity taker cannot buy and sell at the same price. The market maker extracts a "liquidity premium" for providing liquidity.

When you buy with a market order, you pay the ask price. When you sell with a market order, you get the offer price.

(OTOH, what you mean by "real rate" is not at all clear.)

So, in your hypothetical example, assuming all else constant---yes, after making the round trip trade, you will end up with less than you started. This already happens at, for example, the currency exchange booths at an airport. The booth is acting as the market maker.

Now you can also buy with a limit order. That is, post a offer price and wait. If your order gets hit by someone's sell market order, then you get to buy cheap. Downside is execution risk---you may have to wait a while, possibly a very long while.

Some further comments:


The mid-quote between the best bid and ask could be a vacuous quantity when the spread is equal to the minimal price increment. 
The spread is the difference between the best bid and best ask. If an market order size is larger than number of limit orders at those best prices, it will deplete those orders at the best prices and hit orders at the second best price and so on. So if an order size is large or market is thin at the best prices, the spread is a conservative indication of transaction cost. Professional investors split their large orders into smaller ones in order to minimize price impact.
How much the spread matters depends on the type of trader. For a retail trader who occasionally transacts small orders, the cost is probably negligible. On the other hand, slippage is an important issue for institutional investors. Also, on inter-dealer platforms where orders are placed in units of million, spread is closely watched.