I'm asking for the curve construction of the discount curve in the case where payment currency and collateral currency are different. If I refer to BBG, in the case of a USD swap collateralized in EUR, they use the curve N°400 (MBB EUR Coll For USD). How do they construct this curve ?
TO answer the question in the comment. Suppose you have a USD cash flow receivable in 5yrs and you are trying to calculate the PV. You need to know the interest rate that you are paying on the EUR cash collateral. Suppose this is Eonia flat. Then you execute a 5yr currency basis swap where you lend the Euro collateral out at Eonia flat, against borrowing USD on which you pay Fed funds + X, where X is determined by the basis swap market. Then you discount the USD receivable at Fed Funds + X (by which I mean, you construct a new USD curve X bp higher in rate than the standard Fed Funds curve).
If I recall Cooking with collateral by Piterbarg (https://www.risk.net/derivatives/2194249/cooking-collateral) has the details but you effectively need to use FX swaps to get the basis adjust discount rate you want.
In an arbitrage free framework you may be interested to know that a USD cashflow payable in 5y collateralised in EUR can either be valued (in USD) by discounting the cashflows with a USD-EUR discount curve (that is a discount curve for discounting USD cashflows under a EUR CSA).
Or, you can convert the USD cashflow to EUR with the 5y EURUSD FX forward rate, then discount the converted EUR cashflow to the present day with the EUR-EUR discount curve (that is the discount curve for EUR cashflows under a EUR CSA) and then convert the resultant EUR PV to USD with the immediate EURUSD FX rate.
The EURUSD FX Forward rate is essentially determined from the USD-EUR discount curve or vice versa within an arbitrage free framework, so this may practically not be helpful, but it may help with the concept.