The righteousness of God revealed in the gospel (Rom 1:16-17) that was established and brought in by the Lord Jesus Christ (Dan 9:24) is something that is earned, or merited by Him to be legally transferred by the means of imputation to the account of His people by God. It is something that cannot be seen with the physical eye (by sight). In other words, it is invisible. Does this mean it is not actually real? It does require eyes of faith to see (understand) and to believe it is enough (Rom 9:30-33). Some religious folks say it is not enough and is merely a "paste on" or pretend righteousness. This is what the Roman Catholics and other religious folks mean when they call it "Legal Fiction".
The righteousness of God revealed in the gospel, the everlasting righteousness Christ brought in, is as real as it gets. It is the MOST real and legitimate thing in the world. In fact, one of His names proves it so; "The Lord Our Righteousness" (Jer 23:6). The problem most have is that they cannot see it, because it takes the Spirit of God and God-given faith to seeor discern spiritual things. The blindness of the natural, self righteous, defiled conscience must be defeated by the power of God's Spirit through the means of the power of God unto salvation. That powerful action of God, through the glory of His TRUTH, gives life and faith to see, so His people can turn from idols to serve the Living God.
The blessed imputation of Christ's righteousness is real and actually does have an affect that is based on something of measured value by God Himself. It is the value of the glorious merit of Christ’s Person and work. As a Representative of His elect, He was born under the Law to keep, fulfill, and magnify it along with His sacrificial death on the cross completely satisfying Law and justice for the sins of His people. This was righteousness established, the obedience of One (Rom 2:13 & 5:19). "It is finished!", He cried (John 19:30). The vaule and merit of what Christ earned is real and actual, not make-believe or pretend. We know this by Christ being raised from the grave and where He now sits victorious at the right hand of the Father on High. Think it is not real? God who is Judge says it is and the word of God says "it is God who justifies" (Rom 8:32). He is both a just God and a loving Savior. Do you believe Him? "The just shall live by faith."